does this series converge / diverge

Question

does this series converge/diverge conditionally or absolutly

$\sum_{n=2}^{\infty} (-1)^n \cdot \frac{\sqrt{n}}{(-1)^n + \sqrt{n}} \cdot \sin(\frac{1}{\sqrt{n}})$

i can use the facts that:

$\lim_{x\to0}\frac{\sin x}{x} =1 $;

$\sin(\frac{1}{\sqrt{n}})$ is monotone decreasing

my problem is the alternating $-1$'s in the denominator, i dont know how to deal with that, though the fraction looks similar to an $e$ based limit

Answer 1

Since you know the limit of $\sin(x)/x,$ you know that the terms of your series are like $1/\sqrt{n}$ in absolute value. What does this tells you about absolute convergence? As for conditional convergence, if you could prove that the terms of your series are monotonically decreasing in absolute value, you would be done. Can you?

Answer 2

In fact, this is a bit tricky, since the terms are NOT monotonically decreasing. Absolute convergence (or lack thereof) is quite easy, as indicated in my other answer, but for conditional convergence, the only simple way seems to be:

1. Write $\sin(1/\sqrt{n})$ as $1/\sqrt{n} + O(1/n^{3/2}). $

Since $\sum n^{-3/2}$ converges (absolutely), we can throw away that part, so the absolute convergence of the series is equivalent to the convergence of

$$\sum (-1)^n\frac{1}{((-1)^n + \sqrt{n}}.$$

Now in this series, pair terms (the $n$-th with the $n+1$-st). the sum of two such equals:

$$\frac{2 + \sqrt{n} - \sqrt{n+1}}{(-1)^n + \sqrt{n})((-1)^{n+1} + \sqrt{n+1})}.$$

Now, the numerator converges to $2$ as $n$ grows large, but the numerator approaches $n,$ so the series DOES NOT converge conditionally.