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@ -297,10 +297,10 @@ plot!(f, x0, xn)
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From the graph it appears our value for `f(xn)` will underestimate the actual value of the solution slightly.
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##### Example
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##### Example: the power series method and Euler
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The equation $y'(x) = \sin(x \cdot y)$ is not separable, so need not have an easy solution. The default method will fail. Looking at the available methods with `sympy.classify_ode(𝐞qn, u(x))` shows a power series method which can return a power series *approximation* (a Taylor polynomial). Let's look at comparing an approximate answer given by the Euler method to that one returned by `SymPy`.
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The equation $y'(x) = \sin(x \cdot y)$ is not separable, so need not have an easy solution. The default method will fail. Looking at the available methods with `sympy.classify_ode(𝐞qn, u(x))` shows a power series method which can return a power series *approximation* (a polynomial). Let's look at comparing an approximate answer given by the Euler method to that one returned by `SymPy`.
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First, the `SymPy` solution:
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@ -335,6 +335,71 @@ plot!(u, linewidth=5)
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We see that the answer found from using a polynomial series matches that of Euler's method for a bit, but as time evolves, the approximate solution given by Euler's method more closely tracks the slope field.
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----
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The [power series method](https://en.wikipedia.org/wiki/Power_series_solution_of_differential_equations) to solve a differential equation starts with an assumption that the solution can be represented as a power series with some positive radius of convergence. This is formally substituted into the differential equation and derivatives may be taken term by term. The resulting coefficients are equated for like powers giving a system of equations to be solved.
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An example of the method applied to the ODE $f''(x) - 2xf'(x) + \lambda f(x)=0$ is given in the reference above and follows below. Assume $f(x) = \sum_{n=0}^\infty a_n x^n$. Then
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$$
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\begin{align*}
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f(x) &= \sum_{n=0} a_n x^n\\
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f'(x) &= \sum_{n=1} a_n nx^{n-1}\\
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f''(x) &= \sum_{n=2} a_n n (n-1) x^{n-2}\\
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\end{align*}
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$$
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Putting these into the differential equation gives
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$$
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\begin{align*}
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0 &=\sum_{n=2} a_n n (n-1) x^{n-2}
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- 2x \cdot \sum_{n=1} a_n n x^{n-1}
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+ \lambda \sum_{n=0} a_n x^n\\
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&=\sum_{n=0} a_{n+2} (n+2) (n+1) x^{n}
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- \sum_{n=1} 2 n a_n x^{n}
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+ \sum_{n=0} \lambda a_n x^n\\
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&= a_2 (1)(2) + \sum_{n=1} a_{n+2} (n+2) (n+1) x^{n}
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- \sum_{n=1} 2 a_n n x^n
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+ \lambda a_0 + \sum_{n=1} \lambda a_n x^n\\
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&= (\lambda a_0 + 2a_2)\cdot x^0
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+ \sum_{n=1} \left((n+2)(n+1)a_{n+2} + (-2n+\lambda)a_n\right) x^n
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\end{align*}
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$$
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For a power series to be $0$ all its coefficients must be zero. This mandates:
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$$
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\begin{align*}
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0 &= \lambda a_0 + 2a_2,\quad \text{and} \\
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0 &= ((n+2)(n+1)a_{n+2} + (-2n+\lambda)a_n), \quad \text{for } n \geq 1
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\end{align*}
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$$
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The last equation allows one to compute $a_{n+2}$ based on $a_n$. With $a_0$ and $a_1$ parameters we can create the first few values for the terms in the series for the solution:
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```{julia}
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@syms a_0::real a_1::real λ::real
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a_2 = -λ/2 * a_0
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recurse(n) = (2n-λ)/((n+2)*(n+1))
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a_3 = expand(recurse(1)*a_1)
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a_4 = expand(recurse(2)*a_2)
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[a_2, a_3, a_4]
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```
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We can see these terms in the `SymPy` solution which uses the power series method for this differential equation:
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```{julia}
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@syms x::real u()
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∂ = Differential(x)
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eqn = (∂∘∂)(u(x)) - 2*x*∂(u(x)) + λ*u(x) ~ 0
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inits = Dict(u(0) => a_0, ∂(u(x))(0) => a_1)
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dsolve(eqn, u(x); ics=inits)
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```
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##### Example
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@ -801,3 +866,57 @@ choices = [
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answ = 4
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radioq(choices, answ, keep_order=true)
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```
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###### Question
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In the above, we noted that a power series that is always zero must have zero coefficients. Why?
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Suppose we have a series $u(x) = \sum_{n=0} a_n x^n$ with a radius of convergence $r > 0$ such that $u(x) = 0$ for $|x| < r$.
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Why is $u^{n}(x) = 0$ for any $n \geq 0$ and $|x| < r$?
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```{julia}
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#| echo: false
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choices = ["A constant function has derivatives which are constantly zero.",
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"A power series is just a number, hence has derivatives which are always zero."]
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radioq(choices, 1)
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```
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Answer the following as specifically as possible.
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What is the value of $u(0)$?
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```{julia}
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#| echo: false
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choices = [L"0", L"a_0", L"both $0$ and $a_0$"]
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radioq(choices, 3; keep_order=true)
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```
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What is the value of $u'(0)$?
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```{julia}
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#| echo: false
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choices = [L"0", L"a_1", L"both $0$ and $a_1$"]
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radioq(choices, 3; keep_order=true)
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```
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What is the value of $u''(0)$?
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```{julia}
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#| echo: false
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choices = [L"0", L"2a_2", L"both $0$ and $2a_2$"]
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radioq(choices, 3; keep_order=true)
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```
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What is the value of $u^{n}(0)$?
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```{julia}
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#| echo: false
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choices = [L"0", L"n!\cdot a_n", L"both $0$ and $n!\cdot a_n$"]
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radioq(choices, 3; keep_order=true)
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```
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@ -1,4 +1,4 @@
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version: "0.24"
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version: "0.25"
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engines: ['julia']
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project:
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@ -51,6 +51,7 @@ book:
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chapters:
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- limits/limits.qmd
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- limits/limits_extensions.qmd
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- limits/sequences_series.qmd
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- limits/continuity.qmd
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- limits/intermediate_value_theorem.qmd
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@ -1,4 +1,4 @@
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# Taylor polynomials and other approximating polynomials
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# Taylor polynomials, series, and approximating polynomials
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{{< include ../_common_code.qmd >}}
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@ -242,7 +242,6 @@ This immediately applies to the above, where we parameterized by $h$: $x_0=c, x_
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A proof based on Rolle's theorem appears in the appendix.
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## Quadratic approximations; interpolating polynomials
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@ -554,7 +553,7 @@ The output of `series` includes a big "Oh" term, which identifies the scale of t
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:::{.callout-note}
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## Note
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A Taylor polynomial of degree $n$ consists of $n+1$ terms and an error term. The "Taylor series" is an *infinite* collection of terms, the first $n+1$ matching the Taylor polynomial of degree $n$. The fact that series are *infinite* means care must be taken when even talking about their existence, unlike a Tyalor polynomial, which is just a polynomial and exists as long as a sufficient number of derivatives are available.
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A Taylor polynomial of degree $n$ consists of $n+1$ terms and an error term. The "Taylor series" (below) is an *infinite* collection of terms, the first $n+1$ matching the Taylor polynomial of degree $n$. The fact that series are *infinite* means care must be taken when even talking about their existence, unlike a Taylor polynomial, which is just a polynomial and exists as long as a sufficient number of derivatives are available.
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:::
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@ -936,6 +935,46 @@ eqns[2]
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The `solve` function is used to identify $g^{(n)}$ represented in terms of lower-order derivatives of $g$. These values have been computed and stored and are then substituted into `ϕ`. Afterwards a limit is taken and the answer recorded.
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## Taylor series
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Recall a *power series* has the form $\sum_{n=0}^\infty a_n (x-c)^n$. Power series have a radius of convergence ($|x - c| < r$) for which the series converges and diverges when $|x-c| > r$.
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The Taylor polynomial formula can be extended to a formal power series with through
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$$
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a_n = \frac{f^{n}(c)}{n!}.
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$$
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If $f(x)$ is equal to the power series within the radius of convergence derivatives of $f(x)$ can be computed by term-by-term differentiation of the power series. The resulting power series will have the same radius of convergence.
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Consider the Taylor series for $\sin(x)$ and $\cos(x)$ about $0$:
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$$
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\begin{align*}
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\sin(x) &= x - \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots + (-1)^k\frac{x^{2k+1}}{(2k+1)!} + \cdots ...\\
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\cos(x) &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots + (-1)^k\frac{x^{2k}}{(2k)!} + \cdots ...\\
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\end{align*}
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$$
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These both have infinite radius of convergence. Differentiating the power series of $\sin(x)$ term by term gives the power series for $\cos(x)$ as
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$$
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\left[(-1)^k \frac{x^{2k+1}}{(2k+1)!} \right]' =
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(-1)^k \frac{(2k+1) x^{2k}}{(2k+1)!} =
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(-1)^k \frac{x^{2k}}{(2k)!}.
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$$
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Similarly, as the power series for $\sinh(x)$ and $\cosh(x)$ are the same as the above without the alternating signs produced by the $(-1)^k$ term, the term-by-term differentiation of the power series of $\sinh(x)$ produces $\cosh(x)$ and, in this case, vice versa.
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The power series for $e^x$ about $0$ has terms $a_k=x^k/k!$. Differentating gives $kx^{k-1}/k! = x^{k-1}/(k-1)!$. The equivalence of the power series for $e^x$ with its term-by-term differentiation requires a simple shift of indices.
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The power series for $1/(1-x)$ has terms $a_i = x^i$ for $i \geq 0$. The radius of convergence is $1$. Differentiating term-by-term yields a power series for $1/(1-x)^2$ is $a_i = (i+1)x^i$ for $i \geq 0$, which will have a radius of convergence of $1$ as well.
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There are examples (the typical one being $f(x) = e^{-1/x^2}$, defined at $0$ to be $0$) where the function has infinitely many derivatives but the power series and the function are not equal beyond a point. In this example, the function is so flat at $0$ that all its derivatives at $0$ are $0$.
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## Questions
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@ -38,9 +38,11 @@ A function $\vec{f}: R \rightarrow R^n$, $n > 1$ is called a vector-valued funct
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$$
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\vec{f}(t) = \langle \sin(t), 2\cos(t) \rangle, \quad
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\vec{g}(t) = \langle \sin(t), \cos(t), t \rangle, \quad
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\vec{h}(t) = \langle 2, 3 \rangle + t \cdot \langle 1, 2 \rangle.
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\begin{align*}
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\vec{f}(t) &= \langle \sin(t), 2\cos(t) \rangle, \\
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\vec{g}(t) &= \langle \sin(t), \cos(t), t \rangle, \\
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\vec{h}(t) &= \langle 2, 3 \rangle + t \cdot \langle 1, 2 \rangle.\\
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\end{align*}
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$$
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The components themselves are also functions of $t$, in this case univariate functions. Depending on the context, it can be useful to view vector-valued functions as a function that returns a vector, or a vector of the component functions.
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@ -107,21 +109,22 @@ In `Plots`, the command `plot(xs, ys)`, where, say, `xs=[x1, x2, ..., xn]` and `
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Similarly, were a third vector, `zs`, for $z$ components used, `plot(xs, ys, zs)` will make a $3$-dimensional connect the dot plot.
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However, our representation of vector-valued functions naturally generates a vector of points: `[[x1,y1], [x2, y2], ..., [xn, yn]]`, as this comes from broadcasting `f` over some time values. That is, for a collection of time values, `ts` the command `f.(ts)` will produce a vector of points. (Technically a vector of vectors, but points if you identify the $2$-$d$ vectors as points.)
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However, our representation of vector-valued functions naturally generates a vector of points: `[[x1,y1], [x2, y2], ..., [xn, yn]]`, as this comes from broadcasting `f` over some time values. That is, for a collection of time values, `ts` the command `f.(ts)` will produce a vector of vectors. In `Plots`, a vector of *tuples* will be read as a vector of points and plotted accordingly. On the other hand, a vector of vectors is read in as a number of series, with each element being plotted separately. (That is, `[x1,y1]` maps to `plot!([1,2], [x1,y1])`.) To get the desired graph, *either* our function can return a tuple---which makes it clumsier to work with when manipulating the output---or we can turn a vector of points into two vectors---one with the `x` values, one with the `y` values.
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To get the `xs` and `ys` from this is conceptually easy: just iterate over all the points and extract the corresponding component. For example, to get `xs` we would have a command like `[p[1] for p in f.(ts)]`. Similarly, the `ys` would use `p[2]` in place of `p[1]`. The `unzip` function from the `CalculusWithJulia` package does this for us. The name comes from how the `zip` function in base `Julia` takes two vectors and returns a vector of the values paired off. This is the reverse. As previously mentioned, `unzip` uses the `invert` function of the `SplitApplyCombine` package to invert the indexing (the $j$th component of the $i$th point can be referenced by `vs[i][j]` or `invert(vs)[j][i]`).
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To get the `xs` and `ys` from this is conceptually easy: just iterate over all the points and extract the corresponding component. For example, to get `xs` we would have a command like `[p[1] for p in f.(ts)]`. Similarly, the `ys` would use `p[2]` in place of `p[1]`.
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The `unzip` function from the `CalculusWithJulia` package does this for us. The name comes from how the `zip` function in base `Julia` takes two vectors and returns a vector of the values paired off. This is the reverse. As previously mentioned, `unzip` uses the `invert` function of the `SplitApplyCombine` package to invert the indexing (the $j$th component of the $i$th point can be referenced by `vs[i][j]` or `invert(vs)[j][i]`).
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Visually, we have `unzip` performing this reassociation:
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Visually, we have `unzip` performing this re-association:
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```{verbatim}
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[[x1, y1, z1], (⌈x1⌉, ⌈y1⌉, ⌈z1⌉,
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[x2, y2, z2], |x2|, |y2|, |z2|,
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[x3, y3, z3], --> |x3|, |y3|, |z3|,
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[[x₁, y₁, z₁], (⌈x₁⌉, ⌈y₁⌉, ⌈z₁⌉,
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[x₂, y₂, z₂], |x₂|, |y₂|, |z₂|,
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[x₃, y₃, z₃], --> |x₃|, |y₃|, |z₃|,
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⋮ ⋮
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[xn, yn, zn]] ⌊xn⌋, ⌊yn⌋, ⌊zn⌋ )
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[xₙ, yₙ, zₙ]] ⌊xₙ⌋, ⌊yₙ⌋, ⌊zₙ⌋ )
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```
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To turn a collection of vectors into separate arguments for a function, splatting (the `...`) is used.
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@ -175,6 +178,18 @@ ts = range(-2, 2, length=200)
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plot(unzip(h.(ts))...)
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```
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### Using points, not vectors
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As mentioned, there is an alternate manner to plot a vector-valued function that has some conveniences. This is to use a tuple to store the component values. For example:
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```{julia}
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g(t) = (cos(t) + 1/5 * cos(5t), sin(t) + 2/3*sin(3t))
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ts = range(0, 2pi, 251)
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plot(g.(ts); legend=false, aspect_ratio=:equal)
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```
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Broadcasting `g` creates a vector of tuples, which `Plots` treats as points. The drawback to this approach, as mentioned, is that manipulating the output is generally easily when the function output is a vector.
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### The `plot_parametric` function
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@ -229,8 +244,8 @@ For example:
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#| hold: true
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a, ecc = 20, 3/4
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f(t) = a*(1-ecc^2)/(1 + ecc*cos(t)) * [cos(t), sin(t)]
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plot_parametric(0..2pi, f, legend=false)
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scatter!([0],[0], markersize=4)
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plot_parametric(0..2pi, f; legend=false)
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scatter!([(0,0)]; markersize=4)
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```
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@ -272,14 +287,14 @@ function spiro(t; r=2, R=5, rho=0.8*r)
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cent(t) = (R-r) * [cos(t), sin(t)]
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p = plot(legend=false, aspect_ratio=:equal)
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circle!([0,0], R, color=:blue)
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circle!(cent(t), r, color=:black)
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p = plot(; legend=false, aspect_ratio=:equal)
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circle!([0,0], R; linecolor=:blue)
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circle!(cent(t), r; linecolor=:black)
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tp(t) = -R/r * t
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tp(t) = -R / r * t
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s(t) = cent(t) + rho * [cos(tp(t)), sin(tp(t))]
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plot_parametric!(0..t, s, color=:red)
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plot_parametric!(0..t, s; linecolor=:red)
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p
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end
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@ -479,10 +494,15 @@ f(t) = [3cos(t), 2sin(t)]
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t, Δt = pi/4, pi/16
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df = f(t + Δt) - f(t)
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plot(legend=false)
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arrow!([0,0], f(t))
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arrow!([0,0], f(t + Δt))
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arrow!(f(t), df)
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plot(; legend=false, aspect_ratio=:equal)
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plot_parametric!(pi/5..3pi/8, f; line=(:red, 1))
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arrow!([0,0], f(t); line=(:blue,))
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arrow!([0,0], f(t + Δt); line=(:blue,))
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arrow!(f(t), df; line=(:black, 3,0.5))
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annotate!([(f(t)..., text("f(t)", :bottom, :left)),
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(f(t+Δt)..., text("f(t + Δt)", :bottom, :left)),
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((f(t) + df/2)..., text("df", :top, :right)),
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])
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```
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The length of the difference appears to be related to the length of $\Delta t$, in a similar manner as the univariate derivative. The following limit defines the *derivative* of a vector-valued function:
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@ -514,9 +534,12 @@ We can visualize the tangential property through a graph:
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```{julia}
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#| hold: true
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||||
f(t) = [3cos(t), 2sin(t)]
|
||||
p = plot_parametric(0..2pi, f, legend=false, aspect_ratio=:equal)
|
||||
plot(; legend=false, aspect_ratio=:equal)
|
||||
p = plot_parametric!(0..2pi, f; line=(:black, 1))
|
||||
for t in [1,2,3]
|
||||
arrow!(f(t), f'(t)) # add arrow with tail on curve, in direction of derivative
|
||||
arrow!([0,0], f(t); line=(:gray, 1, 0.5))
|
||||
annotate!((2f(t)/3)..., text("f($t)", :top, :left))
|
||||
arrow!(f(t), f'(t); line=(:blue, 2)) # add arrow with tail on curve, in direction of derivative
|
||||
end
|
||||
p
|
||||
```
|
||||
@ -528,8 +551,8 @@ Were symbolic expressions used in place of functions, the vector-valued function
|
||||
|
||||
|
||||
```{julia}
|
||||
@syms 𝒕
|
||||
𝒗vf = [cos(𝒕), sin(𝒕), 𝒕]
|
||||
@syms t
|
||||
vvf = [cos(t), sin(t), t]
|
||||
```
|
||||
|
||||
We will see working with these expressions is not identical to working with a vector-valued function.
|
||||
@ -539,7 +562,7 @@ To plot, we can avail ourselves of the the parametric plot syntax. The following
|
||||
|
||||
|
||||
```{julia}
|
||||
plot(𝒗vf..., 0, 2pi)
|
||||
plot(vvf..., 0, 2pi)
|
||||
```
|
||||
|
||||
The `unzip` usage, as was done above, could be used, but it would be more trouble in this case.
|
||||
@ -549,7 +572,7 @@ To evaluate the function at a given value, say $t=2$, we can use `subs` with bro
|
||||
|
||||
|
||||
```{julia}
|
||||
subs.(𝒗vf, 𝒕=>2)
|
||||
subs.(vvf, t=>2)
|
||||
```
|
||||
|
||||
Limits are performed component by component, and can also be defined by broadcasting, again with the need to adjust the values:
|
||||
@ -557,21 +580,21 @@ Limits are performed component by component, and can also be defined by broadcas
|
||||
|
||||
```{julia}
|
||||
@syms Δ
|
||||
limit.((subs.(𝒗vf, 𝒕 => 𝒕 + Δ) - 𝒗vf) / Δ, Δ => 0)
|
||||
limit.((subs.(vvf, t => t + Δ) - vvf) / Δ, Δ => 0)
|
||||
```
|
||||
|
||||
Derivatives, as was just done through a limit, are a bit more straightforward than evaluation or limit taking, as we won't bump into the shape mismatch when broadcasting:
|
||||
|
||||
|
||||
```{julia}
|
||||
diff.(𝒗vf, 𝒕)
|
||||
diff.(vvf, t)
|
||||
```
|
||||
|
||||
The second derivative, can be found through:
|
||||
|
||||
|
||||
```{julia}
|
||||
diff.(𝒗vf, 𝒕, 𝒕)
|
||||
diff.(vvf, t, t) # or diff.(vvf, t, 2)
|
||||
```
|
||||
|
||||
### Applications of the derivative
|
||||
@ -586,13 +609,13 @@ Here are some sample applications of the derivative.
|
||||
The derivative of a vector-valued function is similar to that of a univariate function, in that it indicates a direction tangent to a curve. The point-slope form offers a straightforward parameterization. We have a point given through the vector-valued function and a direction given by its derivative. (After identifying a vector with its tail at the origin with the point that is the head of the vector.)
|
||||
|
||||
|
||||
With this, the equation is simply $\vec{tl}(t) = \vec{f}(t_0) + \vec{f}'(t_0) \cdot (t - t_0)$, where the dot indicates scalar multiplication.
|
||||
With this, the equation is simply $\vec{tl}(t) = \vec{f}(t_0) + \cdot (t - t_0) \vec{f}'(t_0) $, where the dot indicates scalar multiplication.
|
||||
|
||||
|
||||
##### Example: parabolic motion
|
||||
|
||||
|
||||
In physics, we learn that the equation $F=ma$ can be used to derive a formula for position, when acceleration, $a$, is a constant. The resulting equation of motion is $x = x_0 + v_0t + (1/2) at^2$. Similarly, if $x(t)$ is a vector-valued position vector, and the *second* derivative, $x''(t) =\vec{a}$, a constant, then we have: $x(t) = \vec{x_0} + \vec{v_0}t + (1/2) \vec{a} t^2$.
|
||||
In physics, we learn that the equation $F=ma$ can be used to derive a formula for position, when acceleration, $a$, is a constant. The resulting equation of motion is $x(t) = x_0 + v_0t + (1/2) at^2$. Similarly, if $x(t)$ is a vector-valued position vector, and the *second* derivative, $x''(t) =\vec{a}$, a constant, then we have: $x(t) = \vec{x_0} + \vec{v_0}t + (1/2) \vec{a} t^2$.
|
||||
|
||||
|
||||
For two dimensions, we have the force due to gravity acts downward, only in the $y$ direction. The acceleration is then $\vec{a} = \langle 0, -g \rangle$. If we start at the origin, with initial velocity $\vec{v_0} = \langle 2, 3\rangle$, then we can plot the trajectory until the object returns to ground ($y=0$) as follows:
|
||||
@ -606,7 +629,8 @@ xpos(t) = x0 + v0*t + (1/2)*a*t^2
|
||||
|
||||
t_0 = find_zero(t -> xpos(t)[2], (1/10, 100)) # find when y=0
|
||||
|
||||
plot_parametric(0..t_0, xpos)
|
||||
plot(; legend=false)
|
||||
plot_parametric!(0..t_0, xpos)
|
||||
```
|
||||
|
||||
```{julia}
|
||||
@ -797,7 +821,7 @@ The dot being scalar multiplication by the derivative of the univariate function
|
||||
Vector-valued functions do not have multiplication or division defined for them, so there are no ready analogues of the product and quotient rule. However, the dot product and the cross product produce new functions that may have derivative rules available.
|
||||
|
||||
|
||||
For the dot product, the combination $\vec{f}(t) \cdot \vec{g}(t)$ we have a univariate function of $t$, so we know a derivative is well defined. Can it be represented in terms of the vector-valued functions? In terms of the component functions, we have this calculation specific to $n=2$, but that which can be generalized:
|
||||
For the dot product, the combination $\vec{f}(t) \cdot \vec{g}(t)$ creates a univariate function of $t$, so we know a derivative is well defined. Can it be represented in terms of the vector-valued functions? In terms of the component functions, we have this calculation specific to $n=2$, but that which can be generalized:
|
||||
|
||||
|
||||
$$
|
||||
@ -842,8 +866,8 @@ In summary, these two derivative formulas hold for vector-valued functions $R \r
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
(\vec{u} \cdot \vec{v})' &= \vec{u}' \cdot \vec{v} + \vec{u} \cdot \vec{v}',\\
|
||||
(\vec{u} \times \vec{v})' &= \vec{u}' \times \vec{v} + \vec{u} \times \vec{v}'.
|
||||
(\vec{u} \cdot \vec{v})' &= \vec{u}' \cdot \vec{v} + \vec{u} \cdot \vec{v}', \\
|
||||
(\vec{u} \times \vec{v})' &= \vec{u}' \times \vec{v} + \vec{u} \times \vec{v}'\qaud (n=3).
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
@ -892,7 +916,9 @@ $$
|
||||
\vec{F} = m \vec{a} = m \ddot{\vec{x}}.
|
||||
$$
|
||||
|
||||
Combining, Newton states $\vec{a} = -(GM/r^2) \hat{x}$.
|
||||
(The double dot is notation for two derivatives in a $t$ variable.)
|
||||
|
||||
Combining, Newton's law states $\vec{a} = -(GM/r^2) \hat{x}$.
|
||||
|
||||
|
||||
Now to show the first law. Consider $\vec{x} \times \vec{v}$. It is constant, as:
|
||||
@ -901,7 +927,8 @@ Now to show the first law. Consider $\vec{x} \times \vec{v}$. It is constant, as
|
||||
$$
|
||||
\begin{align*}
|
||||
(\vec{x} \times \vec{v})' &= \vec{x}' \times \vec{v} + \vec{x} \times \vec{v}'\\
|
||||
&= \vec{v} \times \vec{v} + \vec{x} \times \vec{a}.
|
||||
&= \vec{v} \times \vec{v} + \vec{x} \times \vec{a}\\
|
||||
&= 0.
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
@ -1000,7 +1027,7 @@ c^2 &= \|\vec{c}\|^2 \\
|
||||
$$
|
||||
|
||||
|
||||
Solving, this gives the first law. That is, the radial distance is in the form of an ellipse:
|
||||
Solving for $r$, this gives the first law. That is, the radial distance is in the form of an ellipse:
|
||||
|
||||
|
||||
$$
|
||||
@ -1231,23 +1258,25 @@ p
|
||||
---
|
||||
|
||||
|
||||
::: {.callout-note appearance="minimal"}
|
||||
## Curvature of a space curve
|
||||
The *curvature* of a $3$-dimensional space curve is defined by:
|
||||
|
||||
|
||||
> *The curvature*: For a $3-D$ curve the curvature is defined by:
|
||||
>
|
||||
> $\kappa = \frac{\| r'(t) \times r''(t) \|}{\| r'(t) \|^3}.$
|
||||
$$
|
||||
\kappa = \frac{\| r'(t) \times r''(t) \|}{\| r'(t) \|^3}.
|
||||
$$
|
||||
|
||||
|
||||
|
||||
For $2$-dimensional space curves, the same formula applies after embedding a $0$ third component. It can also be expressed directly as
|
||||
For $2$-dimensional space curves, the same formula applies after embedding a $0$ third component. It simplifies to:
|
||||
|
||||
|
||||
$$
|
||||
\kappa = (x'y''-x''y')/\|r'\|^3. \quad (r(t) =\langle x(t), y(t) \rangle)
|
||||
$$
|
||||
|
||||
Curvature can also be defined as derivative of the tangent vector, $\hat{T}$, *when* the curve is parameterized by arc length, a topic still to be taken up. The vector $\vec{r}'(t)$ is the direction of motion, whereas $\vec{r}''(t)$ indicates how fast and in what direction this is changing. For curves with little curve in them, the two will be nearly parallel and the cross product small (reflecting the presence of $\sin(\theta)$ in the definition). For "curvy" curves, $\vec{r}''$ will be in a direction orthogonal of $\vec{r}'$ to the $\sin(\theta)$ term in the cross product will be closer to $1$.
|
||||
:::
|
||||
|
||||
Curvature can also be defined by the derivative of the tangent vector, $\hat{T}$, *when* the curve is parameterized by arc length, a topic still to be taken up. The vector $\vec{r}'(t)$ is the direction of motion, whereas $\vec{r}''(t)$ indicates how fast and in what direction this is changing. For curves with little curve in them, the two will be nearly parallel and the cross product small (reflecting the presence of $\sin(\theta)$ in the definition). For "curvy" curves, $\vec{r}''$ will be in a direction orthogonal of $\vec{r}'$ to the $\sin(\theta)$ term in the cross product will be closer to $1$.
|
||||
|
||||
|
||||
Let $\vec{r}(t) = k \cdot \langle \cos(t), \sin(t), 0 \rangle$. This will have curvature:
|
||||
@ -1269,17 +1298,16 @@ If a curve is imagined to have a tangent "circle" (second order Taylor series ap
|
||||
The [torsion](https://en.wikipedia.org/wiki/Torsion_of_a_curve), $\tau$, of a space curve ($n=3$), is a measure of how sharply the curve is twisting out of the plane of curvature.
|
||||
|
||||
|
||||
::: {.callout-note appearance="minimal}
|
||||
## Torsion of a space curve
|
||||
The torsion is defined for smooth curves by
|
||||
|
||||
|
||||
> *The torsion*:
|
||||
>
|
||||
> $\tau = \frac{(\vec{r}' \times \vec{r}'') \cdot \vec{r}'''}{\|\vec{r}' \times \vec{r}''\|^2}.$
|
||||
|
||||
|
||||
$$
|
||||
\tau = \frac{(\vec{r}' \times \vec{r}'') \cdot \vec{r}'''}{\|\vec{r}' \times \vec{r}''\|^2}.
|
||||
$$
|
||||
|
||||
For the torsion to be defined, the cross product $\vec{r}' \times \vec{r}''$ must be non zero, that is the two must not be parallel or zero.
|
||||
|
||||
:::
|
||||
|
||||
##### Example: Tubular surface
|
||||
|
||||
@ -1294,7 +1322,7 @@ This last example comes from a collection of several [examples](https://github.c
|
||||
The task is to illustrate a space curve, $c(t)$, using a tubular surface. At each time point $t$, assume the curve has tangent, $e_1$; normal, $e_2$; and binormal, $e_3$. (This assumes the defining derivatives exist and are non-zero and the cross product in the torsion is non zero.) The tubular surface is a circle of radius $\epsilon$ in the plane determined by the normal and binormal. This curve would be parameterized by $r(t,u) = c(t) + \epsilon (e_2(t) \cdot \cos(u) + e_3(t) \cdot \sin(u))$ for varying $u$.
|
||||
|
||||
|
||||
The Frenet-Serret equations setup a system of differential equations driven by the curvature and torsion. We use the `DifferentialEquations` package to solve this equation for two specific functions and a given initial condition. The equations when expanded into coordinates become $12$ different equations:
|
||||
The Frenet-Serret equations describe the relationship between tangent, normal, and binormal vectors through a system of differential equations driven by the curvature and torsion. Their derivation will be discussed later, here we give an example usage by using the `DifferentialEquations` package to solve these equations for a specific curvature and torsion function and given initial conditions. The vector equations along with the relationship between the space curve and its tangent vector---when expanded into coordinates---become $12$ different equations:
|
||||
|
||||
|
||||
```{julia}
|
||||
@ -1316,7 +1344,7 @@ function Frenet_eq!(du, u, p, s) #system of ODEs
|
||||
end
|
||||
```
|
||||
|
||||
The last set of equations describe the motion of the spine. It follows from specifying the tangent to the curve is $e_1$, as desired; it is parameterized by arc length, as $\mid c'(t) \mid = 1$.
|
||||
The last set of equations describe the motion of the spine. It follows from specifying the tangent to the curve is $e_1$, as desired (it is parameterized by arc length, as $\lVert c'(t) \rVert = 1$).
|
||||
|
||||
|
||||
Following the example of `@empet`, we define a curvature function and torsion function, the latter a constant:
|
||||
@ -1345,7 +1373,7 @@ prob = ODEProblem(Frenet_eq!, u0, t_span, (κ, τ))
|
||||
sol = solve(prob, Tsit5());
|
||||
```
|
||||
|
||||
The "spine" is the center axis of the tube and is the $10$th, $11$th, and $12$th coordinates:
|
||||
The "spine" is the center axis of the tube and is described by the $10$th, $11$th, and $12$th coordinates:
|
||||
|
||||
|
||||
```{julia}
|
||||
@ -1372,14 +1400,15 @@ ts_0 = range(a_0, b_0, length=251)
|
||||
t_0 = (a_0 + b_0) / 2
|
||||
ϵ = 1/5
|
||||
|
||||
plot_parametric(a_0..b_0, spine)
|
||||
plot(; legend=false)
|
||||
plot_parametric!(a_0..b_0, spine; line=(:black, 2))
|
||||
|
||||
arrow!(spine(t_0), e₁(t_0))
|
||||
arrow!(spine(t_0), e₂(t_0))
|
||||
arrow!(spine(t_0), e₃(t_0))
|
||||
arrow!(spine(t_0), e₁(t_0); line=(:blue,))
|
||||
arrow!(spine(t_0), e₂(t_0); line=(:red,))
|
||||
arrow!(spine(t_0), e₃(t_0); line=(:green,))
|
||||
|
||||
r_0(t, θ) = spine(t) + ϵ * (e₂(t)*cos(θ) + e₃(t)*sin(θ))
|
||||
plot_parametric!(0..2pi, θ -> r_0(t_0, θ))
|
||||
plot_parametric!(0..2pi, θ -> r_0(t_0, θ); line=(:black, 1))
|
||||
```
|
||||
|
||||
The `ϵ` value determines the radius of the tube; we see it above as the radius of the drawn circle. The function `r` for a fixed `t` traces out such a circle centered at a point on the spine. For a fixed `θ`, the function `r` describes a line on the surface of the tube paralleling the spine.
|
||||
@ -1475,7 +1504,7 @@ speed = simplify(norm(diff.(viviani(t, a), t)))
|
||||
integrate(speed, (t, 0, 4*PI))
|
||||
```
|
||||
|
||||
We see that the answer depends linearly on $a$, but otherwise is a constant expressed as an integral. We use `QuadGk` to provide a numeric answer for the case $a=1$:
|
||||
We see that the answer depends linearly on $a$, but otherwise is a constant involving an integral. We use `QuadGk` to provide a numeric answer for the case $a=1$:
|
||||
|
||||
|
||||
```{julia}
|
||||
@ -1573,7 +1602,7 @@ This should be $\kappa \hat{N}$, so we do:
|
||||
```{julia}
|
||||
κₕ = norm(outₕ) |> simplify
|
||||
Normₕ = outₕ / κₕ
|
||||
κₕ, Normₕ
|
||||
κₕ
|
||||
```
|
||||
|
||||
Interpreting, $a$ is the radius of the circle and $b$ how tight the coils are. If $a$ gets much larger than $b$, then the curvature is like $1/a$, just as with a circle. If $b$ gets very big, then the trajectory looks more stretched out and the curvature gets smaller.
|
||||
@ -1778,13 +1807,13 @@ Xₑ(t)= 2 * cos(t)
|
||||
Yₑ(t) = sin(t)
|
||||
rₑ(t) = [Xₑ(t), Yₑ(t)]
|
||||
unit_vec(x) = x / norm(x)
|
||||
plot(legend=false, aspect_ratio=:equal)
|
||||
plot(; legend=false, aspect_ratio=:equal)
|
||||
ts = range(0, 2pi, length=50)
|
||||
for t in ts
|
||||
Pₑ, Vₑ = rₑ(t), unit_vec([-Yₑ'(t), Xₑ'(t)])
|
||||
plot_parametric!(-4..4, x -> Pₑ + x*Vₑ)
|
||||
plot_parametric!(-4..4, x -> Pₑ + x*Vₑ; line=(:black, 1))
|
||||
end
|
||||
plot!(Xₑ, Yₑ, 0, 2pi, linewidth=5)
|
||||
plot!(Xₑ, Yₑ, 0, 2pi, line=(:red, 5))
|
||||
```
|
||||
|
||||
is that of an ellipse with many *normal* lines drawn to it. The normal lines appear to intersect in a somewhat diamond-shaped curve. This curve is the evolute of the ellipse. We can characterize this using the language of planar curves.
|
||||
@ -1858,8 +1887,10 @@ Tangent(r, t) = unit_vec(r'(t))
|
||||
Normal(r, t) = unit_vec((𝒕 -> Tangent(r, 𝒕))'(t))
|
||||
curvature(r, t) = norm(r'(t) × r''(t) ) / norm(r'(t))^3
|
||||
|
||||
plot_parametric(0..2pi, t -> rₑ₃(t)[1:2], legend=false, aspect_ratio=:equal)
|
||||
plot_parametric!(0..2pi, t -> (rₑ₃(t) + Normal(rₑ₃, t)/curvature(rₑ₃, t))[1:2])
|
||||
plot(; legend=false, aspect_ratio=:equal, xlims=(-6,6), ylims=(-5,5))
|
||||
plot_parametric!(0..2pi, t -> rₑ₃(t)[1:2]; line=(:red,5))
|
||||
plot_parametric!(0..2pi, t -> (rₑ₃(t) + Normal(rₑ₃, t)/curvature(rₑ₃, t))[1:2];
|
||||
line=(:black, 1))
|
||||
```
|
||||
|
||||
We computed the above illustration using $3$ dimensions (hence the use of `[1:2]...`) as the curvature formula is easier to express. Recall, the curvature also appears in the [Frenet-Serret](https://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas) formulas: $d\hat{T}/ds = \kappa \hat{N}$ and $d\hat{N}/ds = -\kappa \hat{T}+ \tau \hat{B}$. In a planar curve, as under consideration, the binormal is $\vec{0}$. This allows the computation of $\vec\beta(s)'$:
|
||||
@ -1877,6 +1908,7 @@ $$
|
||||
|
||||
We see $\vec\beta'$ is zero (the curve is non-regular) when $\kappa'(s) = 0$. The curvature changes from increasing to decreasing, or vice versa at each of the $4$ crossings of the major and minor axes--there are $4$ non-regular points, and we see $4$ cusps in the evolute.
|
||||
|
||||
----
|
||||
|
||||
The curve parameterized by $\vec{r}(t) = 2(1 - \cos(t)) \langle \cos(t), \sin(t)\rangle$ over $[0,2\pi]$ is cardiod. It is formed by rolling a circle of radius $r$ around another similar sized circle. The following graphically shows the evolute is a smaller cardiod (one-third the size). For fun, the evolute of the evolute is drawn:
|
||||
|
||||
@ -1891,10 +1923,10 @@ end
|
||||
#| hold: trie
|
||||
r(t) = 2*(1 - cos(t)) * [cos(t), sin(t), 0]
|
||||
|
||||
plot(legend=false, aspect_ratio=:equal)
|
||||
plot_parametric!(0..2pi, t -> r(t)[1:2])
|
||||
plot_parametric!(0..2pi, t -> evolute(r)(t)[1:2])
|
||||
plot_parametric!(0..2pi, t -> ((evolute∘evolute)(r)(t))[1:2])
|
||||
plot(; legend=false, aspect_ratio=:equal)
|
||||
plot_parametric!(0..2pi, t -> r(t)[1:2]; line=(:black, 1))
|
||||
plot_parametric!(0..2pi, t -> evolute(r)(t)[1:2]; line=(:red, 1))
|
||||
plot_parametric!(0..2pi, t -> ((evolute∘evolute)(r)(t))[1:2]; line=(:blue,1))
|
||||
```
|
||||
|
||||
---
|
||||
@ -1911,9 +1943,10 @@ a = t1
|
||||
|
||||
beta(r, t) = r(t) - Tangent(r, t) * quadgk(t -> norm(r'(t)), a, t)[1]
|
||||
|
||||
p = plot_parametric(-2..2, r, legend=false)
|
||||
plot_parametric!(t0..t1, t -> beta(r, t))
|
||||
for t in range(t0,-0.2, length=4)
|
||||
p = plot(; legend=false)
|
||||
plot_parametric!(-2..2, r; line=(:black, 1))
|
||||
plot_parametric!(t0..t1, t -> beta(r, t); line=(:red,1))
|
||||
for t in range(t0, -0.2, length=4)
|
||||
arrow!(r(t), -Tangent(r, t) * quadgk(t -> norm(r'(t)), a, t)[1])
|
||||
scatter!(unzip([r(t)])...)
|
||||
end
|
||||
@ -1924,14 +1957,15 @@ This lends itself to this mathematical description, if $\vec\gamma(t)$ parameter
|
||||
|
||||
|
||||
$$
|
||||
\vec\beta(t) = \vec\gamma(t) + \left((a - \int_{t_0}^t \| \vec\gamma'(t)\| dt) \hat{T}(t)\right),
|
||||
\vec\beta(t) = \vec\gamma(t) + \left(a - \int_{t_0}^t \| \vec\gamma'(t)\| dt\right) \hat{T}(t),
|
||||
$$
|
||||
|
||||
where $\hat{T}(t) = \vec\gamma'(t)/\|\vec\gamma'(t)\|$ is the unit tangent vector. The above uses two parameters ($a$ and $t_0$), but only one is needed, as there is an obvious redundancy (a point can *also* be expressed by $t$ and the shortened length of string). [Wikipedia](https://en.wikipedia.org/wiki/Involute) uses this definition for $a$ and $t$ values in an interval $[t_0, t_1]$:
|
||||
|
||||
|
||||
$$
|
||||
\vec\beta_a(t) = \vec\gamma(t) - \frac{\vec\gamma'(t)}{\|\vec\gamma'(t)\|}\int_a^t \|\vec\gamma'(t)\| dt.
|
||||
\vec\beta_a(t) = \vec\gamma(t) -
|
||||
\frac{\vec\gamma'(t)}{\|\vec\gamma'(t)\|}\int_a^t \|\vec\gamma'(t)\| dt.
|
||||
$$
|
||||
|
||||
If $\vec\gamma(s)$ is parameterized by arc length, then this simplifies quite a bit, as the unit tangent is just $\vec\gamma'(s)$ and the remaining arc length just $(s-a)$:
|
||||
@ -1989,7 +2023,7 @@ $$
|
||||
$$
|
||||
|
||||
|
||||
That is the evolute of an involute of $\vec\gamma(s)$ is $\vec\gamma(s)$.
|
||||
That is, the evolute of an involute of $\vec\gamma(s)$ is $\vec\gamma(s)$.
|
||||
|
||||
|
||||
We have:
|
||||
@ -2041,8 +2075,9 @@ speed = 2sin(t/2)
|
||||
|
||||
ex = r(t) - rp/speed * integrate(speed, t)
|
||||
|
||||
plot_parametric(0..4pi, r, legend=false)
|
||||
plot_parametric!(0..4pi, u -> float.(subs.(ex, t .=> u)))
|
||||
plot(; legend=false)
|
||||
plot_parametric!(0..4pi, r; line=(:black, 1))
|
||||
plot_parametric!(0..4pi, u -> float(subs.(ex, t => u)); line=(:blue, 1))
|
||||
```
|
||||
|
||||
The expression `ex` is secretly `[t + sin(t), 3 + cos(t)]`, another cycloid.
|
||||
|
||||
@ -421,6 +421,125 @@ We want to just say $F'(x)= e^{-x}$ so $f(x) = e^{-x}$. But some care is needed.
|
||||
Finally, at $x=0$ we have an issue, as $F'(0)$ does not exist. The left limit of the secant line approximation is $0$, the right limit of the secant line approximation is $1$. So, we can take $f(x) = e^{-x}$ for $x > 0$ and $0$ otherwise, noting that redefining $f(x)$ at a point will not effect the integral as long as the point is finite.
|
||||
|
||||
|
||||
## Application to series
|
||||
|
||||
|
||||
In this application, we compare a series to a related integral to decide convergence or divergence of the series.
|
||||
|
||||
:::{.callout-note appearance="minimal"}
|
||||
#### The integral test
|
||||
Consider a continuous, monotone decreasing function $f(x)$ defined on some interval of the form $[N,\infty)$. Let $a_n = f(n)$ and $s_n = \sum_{k=N}^n a_n$.
|
||||
|
||||
* If $\int_N^\infty f(x) dx < \infty$ then the partial sums converge.
|
||||
* If $\int_N^\infty f(x) dx = \infty$ then the partial sums diverge.
|
||||
:::
|
||||
|
||||
By the monotone nature of $f(x)$, we have on any interval of the type $[i, i+1)$ for $i$ an integer, that $f(i) \geq f(x) \geq f(i+1)$ when $x$ is in the interval. For integrals, this leads to
|
||||
|
||||
$$
|
||||
f(i) = \int_i^{i+1} f(i) dx
|
||||
\geq \int_i^{i+1} f(x) dx
|
||||
\geq \int_i^{i+1} f(i+1) dx = f(i+1)
|
||||
$$
|
||||
|
||||
Now if $N$ is an integer we have
|
||||
|
||||
$$
|
||||
\int_N^\infty f(x) dx = \sum_{i=N}^\infty \int_i^{i+1} f(x) dx
|
||||
$$
|
||||
|
||||
This translates to this bound
|
||||
|
||||
$$
|
||||
\sum_{i=N}^\infty f(i)
|
||||
\leq \int_N^\infty f(x) dx
|
||||
\leq \sum_{i=N}^\infty f(i+1) = \sum_{i=N+1}^\infty f(i)
|
||||
$$
|
||||
|
||||
If the integral converges, the first inequality implies the series converges; if the integral diverges, the second inequality implies the series diverges.
|
||||
|
||||
### Example
|
||||
|
||||
The $p$-series test is an immediate consequence, as the integral
|
||||
|
||||
$$
|
||||
\int_1^\infty \frac{1}{x^p} dx
|
||||
$$
|
||||
|
||||
converges when $p>1$ and diverges when $p \leq 1$.
|
||||
|
||||
### Example
|
||||
Let
|
||||
|
||||
$$
|
||||
a_n = \frac{1}{n \cdot \ln(n) \cdot \ln(\ln(n))^2}
|
||||
$$
|
||||
|
||||
Does $\sum a_n$ *converge*?
|
||||
|
||||
We use `SymPy` to integrate:
|
||||
|
||||
```{julia}
|
||||
@syms x::real
|
||||
f(x) = 1 / (x * log(x) * log(log(x))^2)
|
||||
integrate(f(x), (x, 3^2, oo))
|
||||
```
|
||||
|
||||
That this is finite shows the series converges.
|
||||
|
||||
---
|
||||
|
||||
The integral of a power series can be computed easily for some $x$:
|
||||
|
||||
:::{.callout-note appearance="minimal"}
|
||||
### The integral of a power series
|
||||
|
||||
Suppose $f(x) = \sum_n a_n (x-c)^n$ is a power series about $x=c$ with radius of convergence $r > 0$. [Then](https://en.wikipedia.org/wiki/Power_series#Differentiation_and_integration) the limits of the integral and the sum can be switched around when $x$ is within the radius of convergence:
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\int f(x) dx
|
||||
&= \int \sum_n a_n(x-c)^n dx\\
|
||||
&= \sum_{n=0}^\infty \int a_n(x-c)^n dx\\
|
||||
= \sum_{n=0}^\infty a_n \frac{(x-c)^{n+1}}{n+1}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
The radius of convergence of this new power series is also $r$.
|
||||
:::
|
||||
|
||||
|
||||
This geometric series has a well known value ($|r| < 1$):
|
||||
|
||||
$$
|
||||
\frac{1}{1-r} = 1 + r + r^2 + \cdots
|
||||
$$
|
||||
|
||||
This gives rise to
|
||||
|
||||
$$
|
||||
\ln(1 - r) = -(r + r^2/2 + r^3/3 + \cdots).
|
||||
$$
|
||||
|
||||
The power series for $e^x$ is
|
||||
|
||||
$$
|
||||
e^x = \sum_{n=0}^\infty \frac{x^n}{n!}
|
||||
$$
|
||||
|
||||
This has integral then given by
|
||||
|
||||
$$
|
||||
\int e^x dx = \sum_{n=0}^\infty \frac{x^{n+1}}{n+1}\frac{1}{n!}
|
||||
= \sum_{n=0}^\infty \frac{x^{n+1}}{(n+1)!}
|
||||
= \sum_{n=1}^\infty \frac{x^n}{n!}
|
||||
= e^x - 1
|
||||
$$
|
||||
|
||||
The $-1$ is just a constant so the antiderivative of $e^x$ is $e^x$.
|
||||
|
||||
|
||||
|
||||
## Questions
|
||||
|
||||
|
||||
|
||||
BIN
quarto/limits/figures/mona-lisa-recursive.png
Normal file
BIN
quarto/limits/figures/mona-lisa-recursive.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 438 KiB |
@ -461,16 +461,6 @@ $$
|
||||
That ${n \choose k} \leq n^k$ can be viewed as the left side counts the number of combinations of $k$ choices from $n$ distinct items, which is less than the number of permutations of $k$ choices, which is less than the number of choices of $k$ items from $n$ distinct ones without replacement – what $n^k$ counts.
|
||||
|
||||
|
||||
### Some limit theorems for sequences
|
||||
|
||||
|
||||
The limit discussion first defined limits of scalar univariate functions at a point $c$ and then added generalizations. The pedagogical approach can be reversed by starting the discussion with limits of sequences and then generalizing from there. This approach relies on a few theorems to be gathered along the way that are mentioned here for the curious reader:
|
||||
|
||||
|
||||
* Convergent sequences are bounded.
|
||||
* All *bounded* monotone sequences converge.
|
||||
* Every bounded sequence has a convergent subsequence. (Bolzano-Weierstrass)
|
||||
* The limit of $f$ at $c$ exists and equals $L$ if and only if for *every* sequence $x_n$ in the domain of $f$ converging to $c$ the sequence $s_n = f(x_n)$ converges to $L$.
|
||||
|
||||
|
||||
## Summary
|
||||
|
||||
1178
quarto/limits/sequences_series.qmd
Normal file
1178
quarto/limits/sequences_series.qmd
Normal file
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