Series & Convergence

An infinite series is the sum of infinitely many terms: $\displaystyle\sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$

To define this rigorously, form the partial sums $S_N = \sum_{n=1}^N a_n$. The series converges to $L$ if $S_N \to L$ as $N \to \infty$; otherwise it diverges.

Geometric series: $\displaystyle\sum_{n=0}^\infty r^n = \frac{1}{1-r}$ for $|r| < 1$; diverges for $|r| \geq 1$.

The $n$-th term test: if $\displaystyle\lim_{n\to\infty} a_n \neq 0$, the series diverges. (The converse is false: $a_n \to 0$ does not guarantee convergence.)