How to decide the convergence or divergence of these series?

How to decide the convergence or divergence of these series?

What about the convergence of the following series?
$$\sum_{n=1}^{\infty} (-1)^n (\frac{1 \cdot 3 \cdot 5 \cdots
(2n-1)}{2\cdot 4 \cdot 6 \cdots (2n)})^3$$
$$\sum_{n=1}^{\infty} (-1)^n (\frac{2n + 100 }{3n + 1 })^n $$
$$\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n} + (-1)^n} $$
$$ \sum_{n=1}^{\infty} \frac{1}{\log ( e^n + e^{-n} ) } $$
$$ \sum_{n=1}^{\infty} (-1)^n \int_{n}^{n+1} \frac{e^{-x}}{x} \ dx $$
$$ \sum_{n=1}^{\infty} \log ( n \sin \frac{1}{n} ) $$
$$ \sum_{n=1}^{\infty} (-1)^n ( 1 - n \sin \frac{1}{n} ) $$
$$ \sum_{n=1}^{\infty} (1 - \cos \frac{1}{n} ) $$
$$ \sum_{n=1}^{\infty} \arctan \frac{1}{2n + 1} $$
$$ \sum_{n=1}^{\infty} (\frac{\pi}{2} - \arctan ( \log n ) ) $$
$$ \sum_{n=1}^{\infty} \sin ( n \pi + \frac{1}{\log n } ) $$
In the case of the second one of our series, is it true that the general
term does not approach zero and so the series is divergent?