Find a convergence interval (radius is 0)/Is a series with 0 sum convergent?

[Mampac]

Hey there,

I'm asked to determine the region on which the series converges, and check on both endpoints.

I started computing the radius using the General Root test, and it turned out to be 0. That means the series converges at one point (0).

The thing is, when I wanna check on the point 0, and I substitute 0 for x in the series, I get 0 as each term of the series, meaning the sum of the series is 0.

What can I say about the series now? Is it convergent at x = 0?

This raises another question: if a series has all zero-elemens and the sum is 0, is it convergent?

Thank you in advance!

 

[Steven G]

How can you test for convergence not depend on xⁿ???!!!!

Are you saying that you do not know the sum of 0+0+0+0+....?

If the sum of a series is a finite sum, then the series is said to converge.