Convergent series

A series $\sum_{n=1}^\infty a_n$ (with $a_n\in\mathbb{R}$ or $\mathbb{C}$) is convergent if its partial sums

converge to a limit $s$ as $N\to\infty$. In that case one writes

Convergent series provide a rigorous meaning to infinite sums and are essential for analytic expansions and approximation. See also absolute convergence and conditional convergence .

Examples:

• The geometric series $\sum_{n=0}^\infty r^n$ converges to $\frac{1}{1-r}$ if $|r|<1$.
• $\sum_{n=1}^\infty \frac{1}{n^2}$ converges (to $\pi^2/6$, though that value is not needed for the definition).
• $\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}$ converges (alternating harmonic series).