Power Series and Radius of Convergence

By the end of this lesson, you will be able to find the interval and radius of convergence for a power series, and differentiate and integrate power series.

Power Series

A power series centered at $a$:

$$\sum_{n=0}^\infty c_n(x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots$$

For each $x$, this is an ordinary series. The key question: for which values of $x$ does it converge?

Radius of Convergence

Apply the Ratio Test with $a_n = c_n(x-a)^n$:

$$\rho = \lim_{n\to\infty}\left|\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}\right| = |x-a|\lim_{n\to\infty}\left|\frac{c_{n+1}}{c_n}\right|$$

The series converges absolutely when $\rho < 1$, i.e., when $|x-a| < R$ where:

$$R = \lim_{n\to\infty}\left|\frac{c_n}{c_{n+1}}\right| \quad \text{(Radius of Convergence)}$$

Three cases:

• $R = 0$: converges only at $x = a$
• $R = \infty$: converges for all $x$ (entire real line)
• $R$ finite: converges on $(a-R, a+R)$; diverges outside; endpoints need separate testing

Finding the Interval of Convergence

1. Use Ratio Test to find $R$ and determine $|x-a| < R$
2. Test each endpoint $x = a-R$ and $x = a+R$ separately
3. The interval of convergence is one of: $(a-R, a+R)$, $[a-R, a+R)$, $(a-R, a+R]$, $[a-R, a+R]$

Example: $\sum_{n=0}^\infty \frac{x^n}{n+1}$

$\rho = \lim\left|\frac{x^{n+1}/(n+2)}{x^n/(n+1)}\right| = |x|\lim\frac{n+1}{n+2} = |x|$

Converges for $|x| < 1$; $R = 1$.

$x = 1$: $\sum 1/(n+1)$ diverges (harmonic-like).

$x = -1$: $\sum (-1)^n/(n+1)$ converges (AST).

Interval of convergence: $[-1, 1)$.

Differentiating and Integrating Power Series

Within the radius of convergence, you can differentiate and integrate term-by-term:

$$\frac{d}{dx}\sum_{n=0}^\infty c_n x^n = \sum_{n=1}^\infty nc_n x^{n-1}$$

$$\int\sum_{n=0}^\infty c_n x^n, dx = \sum_{n=0}^\infty \frac{c_n x^{n+1}}{n+1} + C$$

The radius of convergence stays the same (endpoints may change).

Example: From $\frac{1}{1-x} = \sum_{n=0}^\infty x^n$ (for $|x|<1$):