Conditional and Absolute Convergence

In the discussion on convergence of series I forgot to mention the following two important concepts:

Definition.
(i) A series $\sum_{n=1}^\infty a_n$ is said to converge absolutely if $\sum_{n=1}^\infty |a_n|$ converges.
(ii) A series is said to converge conditionally if $\sum_{n=1}^\infty a_n$ converges but $\sum_{n=1}^\infty |a_n|$ diverges.

In tutorial we have mentioned that if $\sum a_n$ converges absolutely, then $\sum a_n$ converges. By alternating series test there are many series converge conditionally. For example \[
 \sum_{k=1}^\infty (-1)^k \frac{1}{\sqrt{k}}
\] converges by alternating series test, but \[
  \sum_{k=1}^\infty \left|(-1)^k \frac{1}{\sqrt{k}}\right |= \sum_{k=1}^\infty  \frac{1}{\sqrt{k}}
\] diverges by integral test, so $ \sum_{k=1}^\infty (-1)^k \frac{1}{\sqrt{k}}$ converges conditionally.