Divergent series

A divergent series is a series whose partial sums do not converge to a finite limit, i.e. it is not a convergent series .

Divergence may happen because partial sums grow without bound or because they oscillate. A basic necessary condition for convergence is terms go to zero , whose contrapositive often certifies divergence quickly.

Examples:

• The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges.
• The series $\sum_{n=0}^\infty (-1)^n$ diverges, since its partial sums alternate between $1$ and $0$.