Topic 6 of 7

Power series and Taylor series

Step 16 · Calculus 2 topics

Treat power series as functions, find where they converge, and build Taylor polynomials and series that approximate functions.

Power series and radius of convergence

A power series is an endless polynomial, Σ cₙ(x − a)ⁿ. For each x it is a series of numbers. It converges inside an interval centered at a, of radius R, and diverges outside it.

Example: The geometric series

For which x does $\sum_{n=0}^{\infty} x^{n}$ converge, and to what?

It is geometric with ratio x.
It converges when |x| < 1.

AnswerTo $\frac{1}{1 - x}$ , for −1 < x < 1

Example: The ratio test for R

Find the radius of convergence of $\sum \frac{x^{n}}{3^{n}}$ .

The ratio of neighboring terms is $\frac{x}{3}$ .
It converges when |x| ÷ 3 < 1, that is, |x| < 3.

AnswerR = 3

Taylor polynomials

The Taylor polynomial of degree n at a matches f's value and first n derivatives at a. Near a it is a good approximation, and it gets better as n grows.

P_{n}(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^{k}

Example: eˣ at 0

Find the degree 3 Taylor polynomial of eˣ at 0, and use it to estimate e to the power 0.1.

Every derivative of eˣ is eˣ, which is 1 at 0.
P₃(x) = 1 + x + x²/2 + x³/6.
P₃(0.1) = 1 + 0.1 + 0.005 + 0.000167.

AnswerAbout 1.105167; the true value is 1.105171

y = eˣ (solid) with its Taylor polynomials 1 + x and 1 + x + x²/2 (dashed). Near 0 they hug the curve; further away they drift off.

Example: cos x at 0

Find the degree 4 Taylor polynomial of cos x at 0.

The derivatives at 0 cycle through 1, 0, −1, 0, 1.
Only the even powers survive.

Answer $1 - \frac{x^{2}}{2} + \frac{x^{4}}{24}$

Taylor series of common functions

Function	Taylor series at 0	Converges for
$e^{x}$	$1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$	all x
$\sin x$	$x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots$	all x
$\cos x$	$1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots$	all x
$\frac{1}{1 - x}$	$1 + x + x^{2} + x^{3} + \cdots$	\|x\| < 1
$\ln(1 + x)$	$x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \cdots$	−1 < x ≤ 1

Example: Building from a known series

Find the Taylor series of $e^{-x^{2}}$ at 0.

Put −x² in place of x in the series for eˣ.

Answer $1 - x^{2} + \frac{x^{4}}{2!} - \frac{x^{6}}{3!} + \cdots$

Example: A limit by series

Find $\lim_{x \to 0} \frac{\sin x - x}{x^{3}}$ .

$\sin x - x = -\frac{x^{3}}{6} + \frac{x^{5}}{120} - \cdots$ .
Divide by x³: $-\frac{1}{6} + \frac{x^{2}}{120} - \cdots$ .

Answer $-\frac{1}{6}$

Try it yourself

Work each one on paper first, then open the answer to check.

Find the radius of convergence of $\sum \frac{x^{n}}{5^{n}}$ .
Show answer
5
Find the radius of convergence of $\sum \frac{x^{n}}{n!}$ .
Show answer
Infinite: it converges for every x
Find the degree 2 Taylor polynomial of ln x at 1.
Show answer
$(x - 1) - \frac{(x - 1)^{2}}{2}$
Find the degree 3 Taylor polynomial of sin x at 0.
Show answer
$x - \frac{x^{3}}{6}$
Use $1 - \frac{x^{2}}{2}$ to estimate cos 0.2.
Show answer
0.98
Write the Taylor series of $\frac{1}{1 + x}$ at 0.
Show answer
$1 - x + x^{2} - x^{3} + \cdots$