convergent and divergent series

Distinguishing series:

A series converges when the limit of the partial sum

\sum_{k = 0}^{\infty} a_{k} = lim_{n \to \infty} \sum_{k = 0}^{n} a_{k}

exists, and the limit is the value of the series.

Otherwise, it is said to diverge.

A series converges absolutely when

\sum_{k = 0}^{\infty} | a_{k} | = L

A convergent series that is not absolutely convergent is conditionally convergent.