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March 7th, 2018, 06:29 PM   #1
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Lightbulb Is for Grandi's series the Cesaro sum is true

For the series 1-1+1-1+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 +(1-1+1-1+1....)=2+4+6+.... + 1-1+1-1+1.....
Let 1-1+1-1....= 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
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March 7th, 2018, 11:27 PM   #2
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Quote:
Originally Posted by Swapnil Chatterjee View Post
For the series 1-1+1-1+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 +(1-1+1-1+1....)=2+4+6+.... + 1-1+1-1+1.....
Let 1-1+1-1....= 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
It's your value of K that's getting in the way. (Well, S = 1 + 2 + 3 +... probably isn't helping, either.) I can show that
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
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March 8th, 2018, 02:26 AM   #3
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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 $1-1+1-1+\ldots$ (although saying $1-1+1-1+\ldots=\frac12$ is problematic), but by the same method $1+0-1+1+0-1+1\dots$ has a different value ($\frac23$ if I remember correctly).
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Last edited by skipjack; March 8th, 2018 at 04:49 AM.
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March 8th, 2018, 04:54 AM   #4
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Quote:
Originally Posted by Swapnil Chatterjee View Post
For the series 1-1+1-1+1..... = -1/2 it can be found by infinite GP formula
That method will give 1/2, not -1/2. As mentioned already, 1/2 is the Cesàro sum of that series.
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