My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum


Thanks Tree2Thanks
  • 1 Post By romsek
  • 1 Post By v8archie
Reply
 
LinkBack Thread Tools Display Modes
March 28th, 2017, 12:48 AM   #1
Senior Member
 
Joined: May 2015
From: Arlington, VA

Posts: 230
Thanks: 23

Math Focus: Number theory
Series sums' terms choice

Consider:

Sum=1-1+1-1+1-1+1-1...=undefined;

Or does Sum=(1-1)+(1-1)+(1-1)+(1-1)...=0

Please give me other simple examples for infinite summation series where choosing the gathering of terms seems to change its ambiguous outcome.
Loren is offline  
 
March 28th, 2017, 11:35 AM   #2
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: CA

Posts: 1,112
Thanks: 580

nothing mysterious about this

an infinite summation converges to a value only if the underlying sequence converges to 0

your underlying sequence here doesn't converge to 0 so anything you might have to say about it's infinite summation is meaningless
Thanks from topsquark
romsek is offline  
March 28th, 2017, 01:32 PM   #3
Senior Member
 
Joined: May 2015
From: Arlington, VA

Posts: 230
Thanks: 23

Math Focus: Number theory
https://en.wikipedia.org/wiki/Grandi's_series

Above is a great link I just found about such "naive" series summations.
Loren is offline  
March 28th, 2017, 02:07 PM   #4
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,441
Thanks: 874

Math Focus: Elementary mathematics and beyond
https://en.wikipedia.org/wiki/Ramanujan_summation
greg1313 is online now  
March 28th, 2017, 02:09 PM   #5
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 6,557
Thanks: 2148

Math Focus: Mainly analysis and algebra
Quote:
Originally Posted by romsek View Post
your underlying sequence here doesn't converge to 0 so anything you might have to say about it's infinite summation is meaningless
This is the "orthodox" view which relies on us defining the sum to be undefined. Modern research is considerably less definite on the subject. There are good reasons to believe that $\frac12$ is a meaningful value of the series.
Thanks from Loren
v8archie is online now  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
choice, series, sums, terms



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Using series manipulations to find the sums of finite series stevewilliams Algebra 1 April 22nd, 2016 01:40 PM
How to convert a series of sums to a function extreme112 Calculus 3 February 1st, 2015 06:28 PM
what are the sum of the first 8 terms of these series? 1love Algebra 8 May 20th, 2012 08:49 PM
Infinite series - finding sums xdeathcorex Calculus 4 August 31st, 2010 11:03 AM
Series involving natural log and sums close to zero peka0027 Real Analysis 3 April 2nd, 2009 05:22 AM





Copyright © 2017 My Math Forum. All rights reserved.