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
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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
 
$$L=\displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{a_{k+1}}{a_k} \right|.$$
 
$$L=\displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{a_{k+1}}{a_k} \right|.$$
 
<strong>Theorem:</strong> <i>(The ratio test)</i>
 
 
<ol>
 
<ol>
<li>If $L<1$, then the series converges absolutely,</li>
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<li>If $L<1$, then the series [[Absolute convergence|converges absolutely]],</li>
<li>if $L>1$, then the series does not converge,</li>
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<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>
<strong>Proof: █</strong>
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<strong>Proof:</strong> <br />
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==References==
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[https://proofwiki.org/wiki/Ratio_Test]
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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 $$L=\displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{a_{k+1}}{a_k} \right|.$$

  1. If $L<1$, then the series converges absolutely,
  2. if $L>1$, then the series diverges,
  3. if $L=1$, then the test is inconclusive.

Proof:

References

[1]