Here is where I got stuck for part A. I know interval of convergence should be [

Question

Here is where I got stuck for part A. I know interval of convergence should be [-1,1) but I just dont know how we come up with a conclusion from the ratio test.

I am looking to find someone to explain me step by step how you get the result. I am new to this topic so this is really important. Thank you in advance :)

Problem statement In the following series x is a real number. In each case use the ratio test, to determine the radius of convergence of the series. Analyze the behavior of the series at the endpoints in order to determine the interval of convergence.

Explanation / Answer

Well you have got, |X| < 1 by ratio test.

=> -1 < X < 1

And by ratio test, we also know for |X| > 1, the series diverges.

Now, we have to check at X = 1 and at X = -1

At X = -1, an = (-1)^n * n / (n^2 + 1)

By alternating Series test(Leibnitz test), the series converges.

At X = 1, an = n / (n^2 + 1),

By p series test, the series diverges.

=> -1 <= X < 1

X = [-1, 1)

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