Prove the if the series converges then the collection additionally converges. Also, give an example to display that the converse is false, i.e., a instance in i m sorry converges however does not.

You are watching: If an converges then an^2 converges

Proof. i think converges. Then we recognize

. Thus, by the meaning of the limit, for every
0" title="Rendered by QuickLaTeX.com" height="11" width="36" style="vertical-align: 0px;"/> over there exists an creature
together that

" title="Rendered by QuickLaTeX.com"/>

The first sum is a finite amount so that converges, and also the 2nd sum converges through comparison through . Hence, converges

Counterexample. The converse is false. Let

. Then,

" title="Rendered by QuickLaTeX.com"/>

converges. However,

" title="Rendered by QuickLaTeX.com"/>

diverges.

### Related

Prove the if ∑ an converges climate ∑ (an)1/2 n-p converges because that p > 1/2
Prove the ∑ 1/an diverges if ∑ an converges

### Point the end an error, questioning a question, offer an different solution (to use Latex form at the height of your comment):Cancel reply

ArchivesArchivesSelect Month June 2016 (12) might 2016 (142) April 2016 (150) in march 2016 (156) February 2016 (145) January 2016 (155) December 2015 (155) November 2015 (150) October 2015 (155) September 2015 (150) august 2015 (155) July 2015 (155) June 2015 (50)

If you are having actually trouble with math proofs a an excellent book to learn from is How to Prove the by Daniel Velleman:

A yes, really awesome book that I very recommend on just how to examine math and also be a math significant is Laura Alcock"s, How to examine as a mathematics Major:

My complete Math publication Recommendations:
Basics: Calculus, direct Algebra, and Proof WritingCore mathematics Subjects.

See more: If You Get Killed Walk It Off, You Get Hurt, Hurt 'Em Back