My Math Forum Series sums' terms choice

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 March 28th, 2017, 12:48 AM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 385 Thanks: 27 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.
 March 28th, 2017, 11:35 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,126 Thanks: 1104 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
 March 28th, 2017, 01:32 PM #3 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 385 Thanks: 27 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: 7,879 Thanks: 1087 Math Focus: Elementary mathematics and beyond
March 28th, 2017, 02:09 PM   #5
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Quote:
 Originally Posted by romsek 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.

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