Difference between revisions of "Ratio test"
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(Created page with "Let $\{a_1,a_2,\ldots\} \subset \mathbb{C}$ and consider the infinite series $\displaystyle\sum_{k=0}^{\infty} a_k.$ Define $$L=\displaystyle\lim_{k \rightarrow \infty} \left|...") |
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<li>if $L=1$, then the test is inconclusive.</li> | <li>if $L=1$, then the test is inconclusive.</li> | ||
</ol> | </ol> | ||
| + | <strong>Proof: █</strong> | ||
Revision as of 22:13, 3 July 2014
Let $\{a_1,a_2,\ldots\} \subset \mathbb{C}$ and consider the infinite series $\displaystyle\sum_{k=0}^{\infty} a_k.$ Define $$L=\displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{a_{k+1}}{a_k} \right|.$$
Theorem: (The ratio test)
- If $L<1$, then the series converges absolutely,
- if $L>1$, then the series does not converge,
- if $L=1$, then the test is inconclusive.
Proof: █
