PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
determining series convergence (Topic)

Consider a series $\ser$ . To determine whether $\ser$ 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.

The root test and the ratio test are direct applications of the comparison test to the geometric series with terms $(|a_n|)^{1/n}$ 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 | get metadata)

View style:

See Also: if $\sum_{k=1}^\infty a_k$ converges then $a_k\to 0$, limit comparison test, convergent series, infinite product of sums $1\!+\!a_i$


Attachments:
non-existence of universal series convergence criterion (Theorem) by pahio
Log in to rate this entry.
(view current ratings)

Cross-references: geometric series, applications, 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 6436 times total.

Classification:
AMS MSC40A05 (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
Style: Expand: Order:
forum policy
Lacks first condition to convergence by neqm on 2004-01-02 17:17:09
If I'm not mistaken, this lacks the first condition to the convergence of a serie: If the serie is convergent then the limit of its main term (the sequence that constitutes the serie) is zero.


--
Nuno Morgadinho
Undergraduate Computer Science Student
Évora University

[ reply | up ]

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