## Related questions with answers

Question

In this exercise, find the radius of convergence of the power series.

$\displaystyle\sum_{n=0}^{\infty}(-1)^n \frac{x^n}{n+1}$

Solution

VerifiedAnswered 2 years ago

Answered 2 years ago

Step 1

1 of 3The given power series:

$\displaystyle\sum_{n=0}^\infin(-1)^n\frac{x^n}{n+1}$

Let $u_n=(-1)^nx^n/(n+1)$, then:

$\begin{aligned} \underset{n\to\infin}{\lim}\bigg|\frac{u_{n+1}}{u_n}\bigg|&=\underset{n\to\infin}{\lim}\frac{\big|\frac{(-1)^{n+1}x^{n+1}}{n+2}\big|}{\big|\frac{(-1)^nx^n}{n+1}\big|} \\ &=\underset{n\to\infin}{\lim}|x|\frac{n+1}{n+2} \end{aligned}$

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