Math 2565, Spring 2020

The post-COVID lockdown power series wrap-up

Sean Fitzpatrick
University of Lethbridge

Warm-Up

Warm-Up

  1. Given the geometric series formula \(\di \sum_{n=0}^\infty x^n = \frac{1}{1-x}\text{,}\) determine a power series representation for \(\di f(x) = \frac{x^4}{(1-x)^3}\text{.}\)

  2. Give an example of a power series whose interval of convergence is \((0,6]\text{.}\)

  3. Use a power series to evaluate the integral \(\di\int_0^2 e^{x^3}\,dx\)

Taylor Series

Taylor polynomials

Recall: given a function \(f\) differentiable at \(c\text{,}\) its linear approximation at \(c\) is given by

\begin{equation*} \ell(x) = f(c)+f'(c)(x-c)\text{.} \end{equation*}

Higher-order approximations are given by Taylor polynomials:

\begin{equation*} P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(c)}{k!}(x-c)^k\text{.} \end{equation*}

If a Taylor polynomial is centred at \(0\text{,}\) we call it a Maclaurin polynomial.

Taylor series

The Taylor series of an infinitely differentiable function \(f(x)\) is given by

\begin{equation*} \sum_{n=0}^\infty \frac{f^{(n)}(c)}{n!}(x-c)^n\text{.} \end{equation*}

When \(c\) is zero, we get the Maclaurin series for \(f\text{.}\)

Equality of series and function

Recall the Lagrange formula for the remainder \(R_n(x)=f(x)-P_n(x)\) obtained when we approximate a function \(f\) by its degree \(n\) Taylor polynomial:

\begin{equation*} R_n(x) = \frac{f^{(n+1)}(t)}{(n+1)!}(x-c)^{n+1}\text{,} \end{equation*}
where \(t\) is some number between \(x\) and \(c\text{.}\)

For many common functions, it's not hard to show that \(\lim\limits_{n\to \infty}R_n(x)=0\) for all \(x\) in the radius of convergence of the Taylor series.

Letting \(n\to\infty\) in the equality \(f(x)=P_n(x)+R_n(x)\text{,}\) we get

\begin{equation*} f(x) = \sum_{n=0}^\infty\frac{f^{(n)}(c)}{n!}(x-c)^n\text{,} \end{equation*}
provided \(x\) is within the interval of convergence of the series, and is such that \(R_n(x)\to 0\text{.}\)

Common Taylor series

Give the Taylor series for:

  1. \(f(x) = e^x\text{,}\) centred at \(0\text{.}\)

  2. \(g(x) = \cos(x)\text{,}\) centred at \(0\text{.}\)

  3. \(h(x)=\ln(x)\text{,}\) centred at \(1\text{.}\)

Give the interval of convergence for each one.

Manipulating power series

We can add, subtract, multiply and divide power series. Substitution works, too. (Sort of.)

In the case of Taylor series, this is equivalent to doing the same to the corresponding functions.

Example:

Determine Taylor series for:

  1. \(\di f(x) = \frac{x^3}{1-x}\)

  2. \(\di g(x) = \frac{\sin(x)}{e^x}\)

  3. \(h(x) = \sec(x)\)

  4. \(s(x) = \sin(x^3)\)

Binomial series

We have \(\di (1+x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k}x^k\text{,}\) for \(\abs{x}\lt 1\text{.}\)

But note the binomial coefficients are a bit different if \(\alpha\) is not a positive integer:

\begin{equation*} \binom{\alpha}{k} = \frac{\alpha(\alpha-1)\cdots (\alpha-k+1)}{k!}\text{.} \end{equation*}

Example:

Find Taylor series for:

  1. \(f(x) = \sqrt{1+2x}\)

  2. \(\di g(x) = \frac{1}{(1+x^2)^2}\)

Creating new functions

Many new functions (not expressible in terms of elementary functions) arise as power series, often as solutions to differential equations.

Example: Bessel functions.

The Bessel functions are named after Friedrich Bessel, who found them as solutions to Kepler's equations. They also show up in problems involving vibrations.

The Bessel function of order zero is given by

\begin{equation*} J_0(x) = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{2^{2n}(n!)^2}\text{.} \end{equation*}

The Bessel function of order one is given by

\begin{equation*} J_1(x) = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{n!(n+1)!2^{2n+1}}\text{.} \end{equation*}

Limits using Taylor series

Example:

Evaluate the limit:

  1. \(\di \lim_{x\to 0}\frac{\cos(x)-1-x^2}{x^4}\)

  2. \(\di\lim_{x\to 0}\frac{\sqrt{1+x}-1-\frac12 x}{x^2}\)

Integrals using Taylor series

We saw earlier that given a power series \(\di \sum_{n=0}^\infty a_x(x-c)^n\text{,}\) we have

\begin{equation*} \int_a^b \left(\sum_{n=0}^\infty a_n(x-c)^n\right)\,dx = \sum_{n=0}^\infty a_n \int_a^b (x-c)^n\,dx\text{.} \end{equation*}
Now that we know Taylor series replacements for many functions, we can use this to integrate.

Example:

Compute the integral \(\int_0^{1} \frac{\sin(x^2)}{x^2}\,dx\text{.}\)

What is the error in your approxmiation if you use the first 5 terms in the sum to estimate the value of the integral?

Requests

Examples by request

I'll try to fit in a few examples here based on what you asked for on the forum. If you've tuned in live you can suggest others in the comments.

Note: I have no questions planned on direction fields for the test.

A population model

Consider the logistic population model \(\di \frac{dP}{dt}=\frac{3}{500}P(93-p)\text{.}\)

  1. For which values of \(P\) is the population increasing? Decreasing?

  2. Find \(P(t)\) given that \(P(0)=48\text{.}\)

A rate in/rate out problem

Suppose a water tank is being pumped out at 3 L / min . The water tank starts at 10 L of clean water. Water with toxic substance is flowing into the tank at 2 L / min , with concentration \(20t\) g / L at time \(t\text{.}\) When the tank is half empty, how many grams of toxic substance are in the tank (assuming perfect mixing)?

Convergence tests

Determine if the series converges:

  1. \(\di \sum_{n=0}^\infty \frac{n}{e^{n^2}}\)

  2. \(\di \sum_{n=0}^\infty \frac{1}{2^n+3n}\)

  3. \(\di\sum_{n=1}^\infty \frac{n^2+1}{\sqrt{n^3+4n+5}}\)

  4. \(\di \sum_{n=1}^\infty\frac{7^k+k}{k!+5}\)