PlanetMath: determining series convergence

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (190)
Orphanage
Unclass'd (1)
Unproven (496)
Corrections (7)
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

determining series convergence

(Topic)

Consider a series $\Sigma a_n$ . To determine whether $\Sigma a_n$ converges or diverges, several tests are available. There is no precise rule indicating which type of test to use with a given series. The more obvious approaches are collected below.

When the terms in $\Sigma a_n$ are positive, there are several possibilities:
- the comparison test,
- the root test (Cauchy's root test),
- the ratio test,
- -test,
- the integral test.
- Raabe's criteria.
The limit comparison test.
The divergence test.
If the series is an alternating series, then the alternating series test may be used.
Abel's test for convergence can be used when terms in $\Sigma a_n$ can be obained as the product of terms of a convergent series with terms of a monotonic convergent sequence.

The root test and the ratio test are direct applications of the comparison test to the geometric series with terms $(\vert a_n\vert)^{1/n}$ and $\frac{a_{n+1}}{a_n}$ , respectively.

For a paper about tests for convergence, please see this article.

"determining series convergence" is owned by CWoo. [ full author list (3) | owner history (2) ]

(view preamble)

See Also: if $\sum_{k=1}^\infty a_k$ converges then $a_k\to 0$ , limit comparison test, convergent series

Attachments:: non-existence of universal series convergence criterion (Theorem) by pahio

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

Cross-references: geometric series, convergent sequence, monotonic, convergent series, product, alternating series, limit comparison test, Raabe's criteria, integral test, ratio test, root test, comparison test, positive, terms, obvious, diverges, converges, series
There is 1 reference to this entry.

This is version 12 of determining series convergence, born on 2003-02-01, modified 2008-04-19.
Object id is 3958, canonical name is DeterminingSeriesConvergence.
Accessed 5554 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.[ View all 12 ]

Discussion

forum policy

Interact

post | correct | update request | add example | add (any)