for this infinite series (-1)^n/n^2
if i use alternating series test to show that sequence does of a^n does not go to 0, then does this mean that this series is diverging
Yes, if you use the alternating series test and show that the sequence of terms does not approach zero, then it implies that the series is diverging.
To apply the alternating series test to the given infinite series (-1)^n/n^2, we need to check two conditions:
1. The terms of the series must alternate in sign: In this case, (-1)^n alternates between -1 and 1 as n changes.
2. The absolute value of each term must decrease as n increases: In this case, the absolute value of each term is given by 1/n^2 and as n increases, 1/n^2 approaches zero.
Since both conditions of the alternating series test are satisfied, we can conclude that the series converges.
However, you mentioned that the sequence of terms, a^n, does not converge to zero. It seems like there might be a misunderstanding here. In the given series, the terms are (-1)^n/n^2, not a^n. The sequence of terms in this series does converge to zero, so the alternating series test would not indicate divergence.
Therefore, if the sequence of terms converges to zero, the alternating series test will show that the series converges.