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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=11+11+11+11...=undefined; Or does Sum=(11)+(11)+(11)+(11)...=0 Please give me other simple examples for infinite summation series where choosing the gathering of terms seems to change its ambiguous outcome. 
March 28th, 2017, 11:35 AM  #2 
Senior Member 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 
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. 
March 28th, 2017, 02:07 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,441 Thanks: 874 Math Focus: Elementary mathematics and beyond  
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  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.


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