Try Magic Notes and save time.Try it free
Try Magic Notes and save timeCrush your year with the magic of personalized studying.Try it free
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

Verified
Step 1
1 of 3

The 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}

## Recommended textbook solutions #### Thomas' Calculus

14th EditionISBN: 9780134438986 (3 more)Christopher E Heil, Joel R. Hass, Maurice D. Weir
10,144 solutions #### Calculus: Early Transcendental Functions

4th EditionISBN: 9780618606245Bruce H. Edwards, Larson, Robert P. Hostetler
12,642 solutions #### Calculus: Early Transcendentals

8th EditionISBN: 9781285741550 (4 more)James Stewart
11,084 solutions #### Calculus: Early Transcendentals

9th EditionISBN: 9781337613927 (3 more)Daniel K. Clegg, James Stewart, Saleem Watson
11,049 solutions