$\{xn\}$ be a sequence of positive real numbers such that $\sum_{n=1}^{\infty} x_n$ converges. which of following are true
- The series $\sum_{n=1}^{\infty} \sqrt{x_nx_{n+1}}$ converges
- $lim_{n\rightarrow \infty}nx_n =0$
- The series $\sum_{n=1}^{\infty} sin^2 x_n $ converges
- $\sum_{n=1}^{\infty} \frac{\sqrt{x_n}}{1+\sqrt{x_n}}$ converges
I think first can be proved by limit comparison test.
For two by divergence test, $lim_{n\rightarrow \infty}x_n =0$, but counter example for the given statement I dont have. 4 by direct comparison convergent.
Best Answer
$$ 0 \leq \sin^2 x_n \leq x_n^2 < x_n $$ (... because $\sin'(x) \leq 1$ and $\sin''(x) < 0$ for $x \in [0,1]$). By the comparison test, since $\sum_{n=1}^\infty x_n$ converges, so does $\sum_{n=1}^\infty \sin^2 x_n$.