PlanetMath: slower divergent series

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (204)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
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

slower divergent series

(Theorem)

Theorem 1 If

$\displaystyle a_1\!+\!a_2\!+\!a_3\!+\cdots$

(1)

is a diverging series with positive terms, then one can always form another diverging series

$\displaystyle s_1\!+\!s_2\!+\!s_3\!+\cdots$

with positive terms such that

$\displaystyle \lim_{n\to\infty}\frac{s_n}{a_n} = 0.$

(2)

Proof. Let $S_n = a_1\!+\!a_2\!+\cdots+\!a_n$ be the $n^\mathrm{th}$ partial sum of (1). Then we have

$\displaystyle a_n = S_n\!-\!S_{n-1} = (\sqrt{S_n}\!+\!\sqrt{S_{n-1}})(\sqrt{S_n}\!-\!\sqrt{S_{n-1}}).$

We set $s_1 := \sqrt{S_1}$ and

$\displaystyle s_n := \frac{a_n}{\sqrt{S_n}\!+\!\sqrt{S_{n-1}}} = \sqrt{S_n}\!-\!\sqrt{S_{n-1}}$

for $n = 2,\,3,\,4,\,\ldots$ Then the terms of the series

$\displaystyle \sum_{n = 1}^{\infty}s_n = \sqrt{S_1}\!+\!\sum_{n = 1}^{\infty}(\sqrt{S_{n+1}}\!-\!\sqrt{S_n})$

apparently are positive. This series is however divergent, because the sum of its

first terms is equal to $\sqrt{S_n}$ which grows without bound along with

since (1) diverges. For this reason we also get the result (2).

Remark. Niels Henrik Abel has presented a simpler example on such series $s_1\!+\!s_2\!+\!s_3\!+\cdots$ :

$\displaystyle 1\!+\!\frac{a_2}{a_1\!+\!a_2}\!+\!\frac{a_3}{a_1\!+\!a_2\!+\!a_3}\! +\!\frac{a_4}{a_1\!+\!a_2\!+\!a_3\!+\!a_4}\!+\cdots$

"slower divergent series" is owned by pahio.

(view preamble)

Attachments:: proof of slower divergent series (Proof) by rspuzio

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

Cross-references: diverges, bound, sum, divergent, partial sum, positive, series, diverging
There is 1 reference to this entry.

This is version 10 of slower divergent series, born on 2005-03-19, modified 2006-09-27.
Object id is 6885, canonical name is SlowerDivergentSeries.
Accessed 1721 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 2 ]

Discussion

forum policy

No messages.

Interact