Difference between revisions of "Ratio test"
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<strong>Theorem: (The ratio test)</strong> Let $\{a_1,a_2,\ldots\} \subset \mathbb{C}$ and consider the infinite series $\displaystyle\sum_{k=0}^{\infty} a_k.$ Define | <strong>Theorem: (The ratio test)</strong> Let $\{a_1,a_2,\ldots\} \subset \mathbb{C}$ and consider the infinite series $\displaystyle\sum_{k=0}^{\infty} a_k.$ Define | ||
Revision as of 21:45, 11 April 2015
Theorem: (The ratio test) Let $\{a_1,a_2,\ldots\} \subset \mathbb{C}$ and consider the infinite series $\displaystyle\sum_{k=0}^{\infty} a_k.$ Define $$L=\displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{a_{k+1}}{a_k} \right|.$$
- If $L<1$, then the series converges absolutely,
- if $L>1$, then the series diverges,
- if $L=1$, then the test is inconclusive.
Proof: █
