Difference between revisions of "Ratio test"
From specialfunctionswiki
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | |||
| − | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <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 | <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 | ||
| Line 10: | Line 8: | ||
</ol> | </ol> | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
| − | <strong>Proof:</strong> █ | + | <strong>Proof:</strong> █ <br /> |
| + | |||
| + | ==References== | ||
| + | [https://proofwiki.org/wiki/Ratio_Test] | ||
</div> | </div> | ||
</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.
