My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Thanks Tree4Thanks
  • 3 Post By SDK
  • 1 Post By skipjack
Reply
 
LinkBack Thread Tools Display Modes
December 17th, 2018, 03:51 PM   #1
Senior Member
 
Joined: Dec 2015
From: somewhere

Posts: 646
Thanks: 92

Convergence proof

How to show that if $\displaystyle \sum_{n=1}^\infty x^2_n $ converges, then $\displaystyle \sum_{n=1}^\infty \frac{x_n}{n} $ also converges?

Last edited by skipjack; December 18th, 2018 at 04:49 AM.
idontknow is offline  
 
December 17th, 2018, 07:26 PM   #2
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 646
Thanks: 411

Math Focus: Dynamical systems, analytic function theory, numerics
For any $n$, you have $(x_n - \frac{1}{n})^2 > 0$ which implies the bound $\frac{2x_n}{n} \leq x_n^2 + \frac{1}{n^2}$.
Thanks from topsquark, v8archie and idontknow
SDK is offline  
December 18th, 2018, 03:33 AM   #3
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,685
Thanks: 2665

Math Focus: Mainly analysis and algebra
For positive ${x_n}$.
v8archie is offline  
December 18th, 2018, 04:48 AM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 20,975
Thanks: 2225

$\displaystyle \left(\left|x_n\right| - \frac1n\right)^2 \geqslant 0 \implies \left|\frac{2x_n}{n}\right| \leqslant x_n^2 + \frac{1}{n^2} \implies \sum_{n=1}^\infty \frac{x_n}{n}$ is absol.utely convergent.
Thanks from idontknow
skipjack is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
convergence, proof



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Proof of Convergence v8archie Real Analysis 1 December 24th, 2013 11:51 AM
Proof of Convergence of a (recursive) sequence MageKnight Real Analysis 1 May 2nd, 2013 06:40 PM
Convergence proof William_33 Applied Math 1 March 4th, 2013 03:19 PM
proof of convergence....please help! jmu123 Real Analysis 3 December 6th, 2010 01:35 PM
Sequence convergence proof zve5 Real Analysis 2 September 24th, 2008 02:32 PM





Copyright © 2019 My Math Forum. All rights reserved.