PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
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
    $\displaystyle \lim_{n\to \infty} \frac{a_n}{b_n}=L$
    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.
(view preamble)

View style:

See Also: determining series convergence


Attachments:
examples for limit comparison test (Example) by alozano
proof of limit comparison test (Proof) by cvalente
Log in to rate this entry.
(view current ratings)

Cross-references: infinite, diverge, converge, finite, limit, numbers, positive, series
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 4023 times total.

Classification:
AMS MSC: 40-00 (Sequences, series, summability :: General reference works )
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)