Sum of $1/[(n + 1)^2] + 1/[(n + 2)^2] + ... + 1/[(2n)^2]$ when n goes to infinity

Question

Could someone please help me understand the steps to solve this problem?

Answer 1

For finite $n$ the sum is $\psi ^{(1)}(n+1)-\psi ^{(1)}(2 n+1)$, where $\psi$ is the polygamma function. Take the limit:

$$\lim\limits_{n \to \infty} \left( \psi ^{(1)}(n+1)-\psi ^{(1)}(2 n+1) \right) = 0.$$

Answer 2

Hint $$\frac{1}{(2n)^2}+ \frac{1}{(2n)^2} + … + \frac{1}{(2n)^2} \leq \frac{1}{(n+1)^2} + \frac{1}{(n+2)^2} + … + \frac{1}{(2n)^2} \leq\frac{1}{n^2} + \frac{1}{n^2}+ … + \frac{1}{n^2}$$

Answer 3

Hint:

$\frac{n}{(2n)^2} \lt s_n \lt \frac{n}{(n+1)^2}$.