|
|
|
|
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.
- 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.
- 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.
- 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.
|
|
(view preamble | get metadata)
Cross-references: infinite, diverge, converge, finite, limit, numbers, positive, series, theorem
There are 3 references to this entry.
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 MSC: | 40-00 (Sequences, series, summability :: General reference works ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
