Divergent series

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

does not converge in $\mathbb{R}$ or $\mathbb{C}$.

Divergence can occur either because $s_N$ grows without bound (e.g., to $+\infty$) or because it oscillates without approaching a single limit.

Examples:

• $\sum_{n=1}^\infty 1$ diverges since $s_N=N\to\infty$.
• $\sum_{n=1}^\infty \frac{1}{n}$ diverges (harmonic series).
• $\sum_{n=1}^\infty (-1)^n$ diverges because its partial sums oscillate between $-1$ and $0$.