My Math Forum  

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

Real Analysis Real Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
June 30th, 2017, 06:23 AM   #1
Newbie
 
Joined: Jan 2016
From: United Kingdom

Posts: 29
Thanks: 0

result about arithmetic of convergent sequences

Hello all,

I believe that, given a collection of increasing sequences whose total sum converges, each individual sequence will also converge. Attached is my short proof. Is it correct?

Regards,

Magnitude
Attached Images
File Type: jpg IMG_2109.jpg (91.6 KB, 6 views)
Magnitude is offline  
 
June 30th, 2017, 06:49 AM   #2
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 6,854
Thanks: 2228

Math Focus: Mainly analysis and algebra
Your first line seems to state that all the sequences diverge.

Comments:
  1. You need only prove the result for two sequences. The result for $n$ sequences follows by induction.
  2. The given result then follows by considering that the sum is always greater than each of the series individually. Thus, if one diverges we use the comparison test with that series and the sum.

Last edited by v8archie; June 30th, 2017 at 07:07 AM.
v8archie is offline  
June 30th, 2017, 07:31 AM   #3
Newbie
 
Joined: Jan 2016
From: United Kingdom

Posts: 29
Thanks: 0

V8archie,

In my first line, I failed to mention the "collection" is a subset of the original bunch. Then I suppose such a collection could diverge, but then their sum diverges and adding all the rest in (which, having not been included in the collection are bounded above) gives that the total diverges, which is a contradiction. Is this sufficient? Meanwhile, I'm trying the induction approach now.

Last edited by skipjack; June 30th, 2017 at 12:48 PM.
Magnitude is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
arithmetic, convergent, result, sequences



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
convergent sequences Proff Real Analysis 4 September 11th, 2013 09:43 AM
Convergent Sequences lesaltersvi Real Analysis 1 February 8th, 2013 02:49 PM
convergent sequences to 1 - how many? ujemny Applied Math 3 January 3rd, 2013 06:39 AM
product of convergent sequences jmu123 Real Analysis 3 December 8th, 2010 05:40 PM
sequences and series: Arithmetic Sequences cindyyo Algebra 2 August 20th, 2008 02:40 AM





Copyright © 2017 My Math Forum. All rights reserved.