Power Series

Power series are one of the most important topics in Calculus BC.
They allow us to represent functions as infinite polynomials and analyze them like series.

What Is a Power Series?

A power series centered at a is an infinite series of the form:

$$\sum^{\infty}_{n=0}c_{n}(x-a)^{n}$$

where:

• $$c_{n}$$​ = coefficients

• a = center

• x = variable

Example

$$\sum^{\infty}_{n=0}\frac{(x-2)^{n}}{3^{n}}$$​

This is a power series centered at a=2.

Radius and Interval of Convergence

A power series either:

• converges for all $x$, or

• converges only within some interval around $a$, or

• converges only at the center.

We find convergence using Ratio Test.

Ratio Test for Power Series

Given:

$$\sum{c_{n}}(x-a)^{n}$$

Compute:

$$L=\lim_{x\to\infty)\left|\frac{c_{n+1}}{c_{n}}\right|$$​​​

Then the series converges when:

$$|x-a| < \frac{1}{L}$$​

This number is the radius of convergence (R):

$$R=\frac{1}{L}$$​

Interval of Convergence (IOC)

After finding R, check:

$$x=a – R and x=a + R$$

These endpoints may converge or diverge, and must be tested manually.

Example

$$\sum\frac{x^{n}}{n}$$​

Ratio test:

$$\lim_{n\to\infty}\left|\frac{x^{n+1}/(n+1)}{x^{n}/n}\right|=|x|\cdot\lim_{n\to\infty}\frac{n}{n+1}=|x|$$

Converges if:

$$|x|<1$$

Now test endpoints:

• $$x=1 : \sum\frac{1}{n} \longrightarrow$$ diverges

• $$x=-1 : \sun\frac{-1^{n}}{n} \longrightarrow$$ converges

Final IOC:

(-1 , 1]

Power Series as Functions

If a power series converges for some $x$, then:

• It represents a function

• It behaves like a polynomial

• We can differentiate and integrate term by term

Differentiation of Power Series

Given:

$$f(x)=\sum^{\infty}_{n=1}nc_{n}(x-a)^{n-1}$$

Differentiate term-by-term:

$$f^\prime(x)=\sum^{\infty}_{n=1}nc^{n}(x-a)^{n-1}$$

The radius of convergence stays the same.

Integration of Power Series

Integrate term-by-term:

$$\int f(x)dx = \sum^{\infty}_{n=0}\frac{c_{n}}{n+1}(x-a)^{n+1}+C$$

Radius of convergence stays the same.