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March 7th, 2018, 06:29 PM  #1 
Newbie Joined: Mar 2018 From: Bardhaman Posts: 1 Thanks: 0  Is for Grandi's series the Cesaro sum is true
For the series 11+11+1..... = 1/2 it can be found by infinite GP formula or by an alternative method too. But let S =1+2+3+4+...... 2S = 2+4+6+8+..... 2S +(11+11+1....)=2+4+6+.... + 11+11+1..... Let 11+11....= K 2S +K = 2(3 + 7 + 11+ ....) 2S +K = 2((1+2)+(3+4)+(5+6)+.....) Rearrange rhs 2S + K = 2(1+2+3+4+5+6+.....) 2S+ K = 2S Hence K = 0 it's a surprise to get this result 
March 7th, 2018, 11:27 PM  #2  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,819 Thanks: 727 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
1  1 + 1  1 + ... = 1 1  1 + 1  1 + ... = 0 1  1 + 1  1 + ... = 1/2 The reason your proof "fails" is that the sum represented by K is not well defined. Dan  
March 8th, 2018, 02:26 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,331 Thanks: 2457 Math Focus: Mainly analysis and algebra 
The OP is just shuffling symbols around without any thought to whether the manipulations make sense. We already know that infinite addition is not associative (or commutative) because we can make any conditionally convergent series sum to any desired value by changing the order of evaluation. It's the "rearrange this" that is invalid. The Cesàro summation is an accepted way to assign a meaningful value to $11+11+\ldots$ (although saying $11+11+\ldots=\frac12$ is problematic), but by the same method $1+01+1+01+1\dots$ has a different value ($\frac23$ if I remember correctly). Last edited by skipjack; March 8th, 2018 at 04:49 AM. 
March 8th, 2018, 04:54 AM  #4  
Global Moderator Joined: Dec 2006 Posts: 19,190 Thanks: 1649  Quote:
 

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cesaro, grandi, series, sum, true 
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