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limit comparison test (Theorem)

The following theorem is a powerful test for convergence of series.

Theorem 1 (Limit Comparison Test)   Let $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ be two series of positive numbers.
  1. If the limit $$\lim_{n\to \infty} \frac{a_n}{b_n}=L$$ exists and $L\neq 0$ is a non-zero finite number, then both series $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ converge or both diverge.
  2. If $L=0$ and $\sum_{n=0}^\infty b_n$ converges then $\sum_{n=0}^\infty a_n$ converges as well. If $L=0$ and $\sum_{n=0}^\infty a_n$ diverges then $\sum_{n=0}^\infty b_n$ diverges as well.
  3. Similarly, if the limit is infinite (``$L=\infty$ '') and $\sum_{n=0}^\infty a_n$ converges then $\sum_{n=0}^\infty b_n$ converges as well. If $L=\infty$ and $\sum_{n=0}^\infty b_n$ diverges then $\sum_{n=0}^\infty a_n$ diverges as well.




"limit comparison test" is owned by alozano.
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See Also: determining series convergence


Attachments:
examples for limit comparison test (Example) by alozano
proof of limit comparison test (Proof) by cvalente
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Cross-references: infinite, diverge, converge, finite, limit, numbers, positive, series, theorem
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This is version 1 of limit comparison test, born on 2005-02-10.
Object id is 6734, canonical name is LimitComparisonTest.
Accessed 5608 times total.

Classification:
AMS MSC40-00 (Sequences, series, summability :: General reference works )

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