# Re: Caesaro summability: all positive integers sum to -1/12

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Posted by Adam Getchell on October 13, 2002 at 03:47:11:

In Reply to: Re: Caesaro summability: all positive integers sum to -1/12 posted by T.Gracken on October 04, 2002 at 07:43:26:

: : Anyone know how to prove the above? That is,

: : (Sum for n = 0 to infinity) n = -1/12

: not so. the sum (for n=0 to infinity) of n is infinity [i.e. diverges] and if you think about fundamentals, you can not have a sum less than zero when all of the addends are non-negative.

: : I can use Cesaro sums to prove that the harmonic series,

: : (Sum for n = 0 to infinity) (-1)^n = 1/2, but I cannot extrapolate to convince myself of the proof above.

: the harmonic series is not represented by what you wrote above. the nth term of the sequence is n-1 and the series (from 1 to infinitiy) diverges (...unless you're Dr C)

You are incorrect.

The sum of the divergent series 1 + 2 + 3 + ... = -1/12 was mentioned in Dr. Berry's article "Singular Limits" in May 2002 Physics Today. The alternating (infinite) divergent series 1+1-1 ... =1/2 can be proved with Cesaro sums.

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