Exercise 332Calculus › Single-variable calculus › Series › Power seriesL2Added 31 Oct 2026

Radius of convergence

#power series#ratio test

Statement

Find the radius of convergence of the power series $\displaystyle\sum_{n\ge1}\frac{n}{2^n}x^n$ , and study its behaviour at the endpoints of the interval of convergence.

Hint

Use d'Alembert's ratio test on $a_n=\dfrac n{2^n}$ .

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Solution

Let $a_n=\dfrac n{2^n}$ . Then
$\frac{a_{n+1}}{a_n}=\frac{n+1}{2n}\longrightarrow\frac12,$
so the radius of convergence is $R=2$ .

At $x=\pm2$ the general term is $n\,(\pm1)^n$ , whose absolute value $n$ does not tend to $0$ . The series diverges at both endpoints, so the interval of convergence is the open interval $(-2,2)$ .

Radius of convergence

Statement

Solution

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