Math 2565, Spring 2020

Taylor Series

March 12, 2020

Sean Fitzpatrick
University of Lethbridge

Introduction and recap

Warm-Up

Find the radius and interval of convergence:

  1. \(\di \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}x^n\)

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

Quiz 10

Identify the function whose Maclaurin series is given by:

  1. \(\di \sum_{n=0}^\infty \frac{x^n}{n!}\)

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

Power Series

Examples

Determine radius and interval of convergence:

  1. \(\di \sum_{n=0}^\infty (-2)^nx^n\)

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

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

  4. \(\di \sum_{n=0}^\infty \frac{(2x)^n}{n!}\)

Derivatives and Integrals

One of the main reasons power series are useful is that derivatives and integrals work the same as they do for polynomials.

Theorem:

Let \(\di f(x)=\sum_{n=0}^\infty a_n(x-c)^n\) be a power series with radius of convergence \(R\text{.}\) Then:

  1. \(\di f'(x) = \sum_{n=1}^\infty na_n(x-c)^{n-1}\text{,}\) with radius of convergence \(R\text{.}\)

  2. \(\di \int f(x)\,dx = C+\sum_{n=0}^\infty \frac{a_n}{n+1}(x-c)^{n+1}\text{,}\) with radius of convergence \(R\text{.}\)

Examples

  1. \(\di\sum_{n=0}^\infty x^n\)

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

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

  4. \(\di \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)!}x^{2n+1}\)

  5. \(\di \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n)!}x^{2n}\)

Initial Value Problems

A common use for power series is finding solutions to differential equations.

We simply suppose that there is a solution \(f(x)=\sum_{n=0}^\infty a_nx^n\text{,}\) and plug it in.

Example:

Use power series to solve the following:

  1. \(y'-2xy=0\text{,}\) with \(y(0)=5\)

  2. \(y''+4y = 0\text{,}\) with \(y(0)=4\) and \(y'(0)=-2\)

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

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