Determine if power series convergence

Question

I am supposed to determine the converge field of:

$$\sum_{n=1}^{\propto }n^{2}x^{n-1}$$ I do not know how to solve it, because I cannot see the path how to use ration criterion, or Cauchy–Hadamard criterion.

I know that the correct asnwer is $\left ( -1,1 \right )$

Can anyone help me?

@AbhasKumarSinha I edited my question, can you look at it one more time? — , Dec 07 '19 at 12:50
Ratio test does give the answer. Where exactly is your difficulty? — Kavi Rama Murthy, Dec 07 '19 at 12:52
https://math.stackexchange.com/questions/593996/how-to-prove-sum-n-0-infty-fracn22n-6 — lab bhattacharjee, Dec 07 '19 at 12:57

score 0 · Answer 1 · answered Dec 07 '19 at 12:54

0

Use the fact that radius of convergence = 1/r Where r = Lim $a_n^{1/n}$

Here that limit is clearly 1. Hence it converges in (-1,1) surely. Now check -1,1 points explicitly.

answered Dec 07 '19 at 12:54

Angry_Math_Person

2,348

Thanks, they do not belong there – Dec 07 '19 at 12:58

Determine if power series convergence

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