Difference between revisions of "Ratio test"
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| − | Let $\{a_1,a_2,\ldots\} \subset \mathbb{C}$ and consider the infinite series $\displaystyle\sum_{k=0}^{\infty} a_k.$ Define | + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> |
| + | <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 | ||
$$L=\displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{a_{k+1}}{a_k} \right|.$$ | $$L=\displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{a_{k+1}}{a_k} \right|.$$ | ||
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| − | |||
<ol> | <ol> | ||
| − | <li>If $L<1$, then the series converges absolutely,</li> | + | <li>If $L<1$, then the series [[Absolute convergence|converges absolutely]],</li> |
| − | <li>if $L>1$, then the series | + | <li>if $L>1$, then the series [[diverge|diverges]],</li> |
<li>if $L=1$, then the test is inconclusive.</li> | <li>if $L=1$, then the test is inconclusive.</li> | ||
</ol> | </ol> | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ <br /> | ||
| + | |||
| + | ==References== | ||
| + | [https://proofwiki.org/wiki/Ratio_Test] | ||
| + | </div> | ||
| + | </div> | ||
Latest revision as of 18:38, 1 December 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.
