PlanetMath: comparison test

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (192)
Orphanage
Unclass'd (1)
Unproven (490)
Corrections (5)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

comparison test

(Theorem)

The series

$\displaystyle \sum_{i=0}^\infty a_i$

with real

is absolutely convergent if there is a sequence $(b_n)_{n\in\mathbb{N}}$ with positive real

such that

$\displaystyle \sum_{i=0}^\infty b_i$

is convergent and for all sufficiently large

holds $\vert a_k\vert\leq b_k$ .

Also, the series $\sum a_i$ is divergent if there is a sequence with positive real , so that $\sum b_i$ is divergent and $a_k\geq b_k$ for all sufficiently large .

"comparison test" is owned by mathwizard.

(view preamble)

Attachments:: proof of comparison test (Proof) by mathwizard

Log in to rate this entry.
(view current ratings)

Cross-references: divergent, positive, sequence, absolutely convergent, real, series
There are 9 references to this entry.

This is version 1 of comparison test, born on 2003-01-10.
Object id is 3889, canonical name is ComparisonTest.
Accessed 3460 times total.

Classification:

AMS MSC:

40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact