Ratio test

Let $\{a_1,a_2,\ldots\} \subset \mathbb{C}$ and consider the infinite series $\displaystyle\sum_{k=0}^{\infty} a_k.$ Define $$L=\displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{a_{k+1}}{a_k} \right|.$$

Theorem: (The ratio test)

1. If $L<1$, then the series converges absolutely,
2. if $L>1$, then the series does not converge,
3. if $L=1$, then the test is inconclusive.

Proof: █