Divergent series

Why do the following series diverge?

Variant 1
\(\displaystyle \sum_{n=1}^{\infty}(-1)^n\ . \)
Resolution
The sequence \(a_n=(-1)^n\) has no limit. Specifically, it is not true that \(\displaystyle \lim_{n\to\infty}a_n=0, \) and so from the necessary conditions for convergence we know that the given series diverges.
Variant 2
\(\displaystyle \sum_{n=1}^{\infty}\frac{n}{\sqrt{n^2+1}} \)
Resolution
\(\displaystyle \lim_{n\to\infty}\frac{n}{\sqrt{n^2+1}}=1 \), so the series cannot converge.