Series

A series is a formal expression $\sum_{n=1}^\infty a_n$ together with the associated partial sums $s_n=\sum_{k=1}^n a_k$.

Whether a series has a value is determined by the behavior of its partial sums: it either converges or diverges . Many standard tools for deciding this are collected as convergence tests such as the ratio test and the comparison test .

Examples:

• The geometric series $\sum_{n=0}^\infty r^n$.
• The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$.