Power Series Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Power Series.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An infinite series of the form \sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots where c is the center and a_n are the coefficients. A power series defines a function of x wherever it converges.

A power series is an 'infinite polynomial' centered at c. For each value of x, you get a number series that may or may not converge. The set of x-values where it converges forms an interval centered at c, and within that interval, the power series behaves like a well-defined function.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Every power series has a radius of convergence R: it converges absolutely for |x-c| < R and diverges for |x-c| > R. At the endpoints x = c \pm R, convergence must be checked individually. Within its interval, a power series can be differentiated and integrated term by term.

Common stuck point: Finding the radius of convergence is usually straightforward (ratio or root test), but checking the endpoints requires separate analysis—often using alternating series test or p-series comparison. Don't forget the endpoints!

Sense of Study hint: Apply the ratio test to |a_(n+1)(x-c)^(n+1) / (a_n(x-c)^n)| and solve for which x-values make the limit less than 1.

Worked Examples

Example 1

medium
Find the radius of convergence of \displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n+1}.

Solution

  1. 1
    Apply the ratio test: compute \left|\frac{a_{n+1}}{a_n}\right| where a_n = \frac{x^n}{n+1}.
  2. 2
    Simplify the ratio: \left|\frac{x^{n+1}}{n+2} \cdot \frac{n+1}{x^n}\right| = |x|\cdot\frac{n+1}{n+2} \xrightarrow{n\to\infty} |x|
  3. 3
    The series converges when L = |x| < 1, so the radius of convergence is R = 1.

Answer

R = 1
The ratio test extracts R by finding the limiting ratio. Here L = |x|, so R=1.

Example 2

hard
Find the interval of convergence of \displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the radius of convergence of \displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!}.

Example 2

medium
Differentiate \sum_{n=0}^{\infty} x^n = \frac{1}{1-x} term by term to find a new series identity.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

convergence divergencetaylor seriessigma notation