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Answered question

2021-12-02

For what values of is each series convergent?

\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1}}{n^{p}}

Answer & Explanation

Kevin Hunt

Beginner2021-12-03Added 20 answers

\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1}}{n^{p}}

This is an alternating series, but for

p \leq 0

we can try the Test for Divergence first to see if we can rule out convergence quickly before trying the Alternating Series Test.
If p<0 then we get

lim_{n \Rightarrow \infty} {(- 1)}^{n - 1} n^{| p |} = D N E

It is oscillating and doesnt

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