Question

For what value of p does \(\large\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n}{n^p}\) converge?

Answer 1

My guess would be \(0<p<\infty\)

Answer 2

Am I correct?

Answer 3

Isn't that just the p-test? p-Series Convergence: http://www.math.com/tables/expansion/tests.htm

if p>1 it converges.

Answer 4

Well, with \((-1)^n\), it's different story. Using Alternating series test, \(\sum (-1)^na_n\) converges if \(\lim_{n \rightarrow \infty}a_n = 0\)

Answer 5

Yeah i was thinking use the alternating series test... too tired though, i should be asleep :P

Answer 6

\(a_n = \dfrac{1}{n^p}\) so I'm sure it's 0 < p < ∞

Answer 7

Oh lol ok.

Answer 8

Yeah it should converge for 0 < p < inf then, since the terms are decreasing.

Answer 9

yeah

Answer 10

Just want to make sure, sorry. Thanks.