Computing Sums of Series

$\sum 3^{n} \sin^{3} \frac{x}{3^{n}}$.

$\sum (-1)^{n} \frac{\cos^{3}\left(3^{n} x\right)}{3^{n}}$.

$\sum \operatorname {arctg} \frac{1}{n^{2}+n+1}$.

$\sum \operatorname {arctg} \frac{1}{2 n^{2}}$.

$\sum_{n \geqslant 1} \frac{1}{C_{n+r}^{1+r}}(r \in \mathbb {N})$.

$\sum_{n \geqslant 0} \frac{(n!)^{2} z^{n}}{(2 n+1)!} \quad (|z|<4)$.

$\sum n(\operatorname {ch} 2 n-e) e^{-n^{2}}$.

$\sum_{n \geqslant 2}\left(\frac{1}{n} \sum_{m \geqslant 2} m^{-n}\right)$.

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

$\sum \frac{(-1)^{n}}{n}\left[\log_{2} n\right]$.

$\sum \left(\frac{1}{2 n-1}-\ln \left(1+\frac{1}{2 n-1}\right)\right)$.

$\sum \left(\frac{1}{2 n}-\ln \left(1+\frac{1}{2 n}\right)\right)$.

$\sum \left(1-\left(n+\frac{1}{2}\right) \ln \left(1+\frac{1}{n}\right)\right)$.

$\sum \frac{1}{2^{n}} \operatorname {tg} \frac{1}{2^{n}}$.

$\sum \frac{1}{n} \int_{2 \pi n}^{\infty } \frac{\sin x}{x} d x$.

$\sum \frac{1}{n} \int_{\pi n}^{\infty } \frac{\sin x}{x} d x$.

Prove that

In particular,

Prove that

Here $p_{1}=2, p_{2}=3, \ldots$ are the prime numbers, numbered in increasing order.

Prove that for $\left|t\right|<1$

Find the sum of the series $\sum_{n \geqslant 1} \frac{a_{n}(m)}{n}$, where $m \in \mathbb {N}, m>1$ and $a_{n}(m)=1$ if $n$ is not divisible by $m$, $a_{n}(m)=1-m$ if $n$ is divisible by $m$.