$\sum\limits_{i=1}^\infty \frac{(-1)^n}{\sqrt n}$ The alternating series test requires that $\frac{1}{\sqrt n }$ be convergent. However by p series test, we know p<1 => $\frac{1}{\sqrt n }$ is divergent. Does this show that the sum is divergent aswell?
$\begingroup$
$\endgroup$
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
You are misunderstanding the alternating series test. It requires that $1/\sqrt n$ be decreasing and $\lim 1/\sqrt n=0$. Because both of these conditions are satisfied, the given series converges.
answered Oct 18, 2015 at 17:12
$\begingroup$
$\endgroup$
The $p$-series test is relevant for series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$. However, you are interested in the convergence of $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ which is not of this form so this test is irrelevant. However, using the alternating series test you can conclude that it does converge.
