what test should I use for this series:

Question

anonymous · Answer

$$\sum_{n=1}^{\infty}\frac{ 2^{n}(\frac{ 1 }{ 2 })^{n} }{ (n+1)^{2} }$$

perl · Answer

the 2^n and 1/2 ^n cancel each other

jhannybean · Answer

$$2^n\cdot \left(\dfrac{1}{2}\right)^n \implies \left(2\cdot \dfrac{1}{2}\right)^n = (1)^n$$

jhannybean · Answer

$$\large \sum_{n=1}^{\infty} \frac{1^n}{(n+1)^2}$$

perl · Answer

1^n = 1

perl · Answer

1^1 = 1^2 = 1^3 = ... 1^n = 1

jhannybean · Answer

Haha yeah.

jhannybean · Answer

OP is not responding, unfortunately.

perl · Answer

there might be a typo in this question, can the OP take a screenshot of the original question

anonymous · Answer

@Jhannybean @perl sorry I was waiting for the responses and started doing other things! I did not see the cancellation!! haha. this is a question from my textbook, so no screenshot. the original question is to find the radius and interval of convergence of the following power series: $$\sum_{n=0}^{\infty}\frac{ 2^{n}x ^{n} }{ (n+1)^{2} }$$ i found the interval of convergence to be -0.5<x<0.5, but I had to check the endpoints. I thought of using the alternating series test which is how I ended up at the question I ask you guys about..I just could not think of a way to simplify the limit.