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Let $S=1+(-1)+1+(-1)+...=\sum_{n=0}^{\infty}(-1)^n$, under which sense does it sum up to 1/2?

I think it shall be one among Euler summation, Borel summation, Cesàro summation, or a subset, but I don't really know these summation methods. Would anyone be kind enough to elaborate them for me, or just point out which summation should it be and I'll look it up in the books?

arax
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  • I am sure this is a duplicate of something – Lost1 Jan 17 '14 at 18:04
  • @Lost1 I got the same feeling and did the googling, but I can't find the proper keyword to use. And I really don't want to learn all about these summations for this simple problem. So I came here for help. – arax Jan 17 '14 at 18:07
  • start from http://math.stackexchange.com/q/635324/ and linked questions – Grigory M Jan 17 '14 at 18:56
  • ...and, of course, there is http://math.stackexchange.com/q/39802/ (and links in comments there) – Grigory M Jan 17 '14 at 19:04

4 Answers4

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The answer is a Cesàro Summation. Check the example on Grandi's Series.

ireallydonknow
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Consider $$ S(x)=\sum_{k=0}^\infty x^k $$ Clearly $xS(x)+1=\sum_{k=0}^\infty x^k=S(x)\implies S(x)=\frac{1}{1-x}$. However, you may note that convergence tests yield that $S$ only converges for $|x|<1$ so it is not accurate to say $S(-1)=\frac{1}{1-(-1)}$. What we can do is say $$ \lim_{x\to -1}\sum_{k=0}^\infty x^k=\frac12 $$

Tim Ratigan
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Formally, $$\frac{1}{1-(-1)}=1-1+1-1+\cdots,$$ and the left hand side is equal to $1/2$, although this is not really valid hence use of the word formal. Note that $$\frac{1}{1-x}=x^0+x^1+x^2+x^3+\cdots,$$ where $|x|<1$.

pshmath0
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Using just the axioms of a summation "machine" that are $$S(\{\alpha a_i+\beta b_i\}_{i\in\mathbb{N}})=\alpha S(a_1,a_2,a_3,...)+\beta S(b_1,b_2,b_3,...)\\S(x_1,x_2,x_3,...)=x_1+S(x_2,x_3,...)$$ we have $S=1-1+1-1+...=1+(-1)\cdot S\Rightarrow 2S=1\Rightarrow S=\frac 1 2$

In order to be precise, before being allowed to use Euler summation, Borel summation, you have to verify that these methods fullfill the axioms of such a machine.

Reference: Carl Bender, Lecture on physical mathematics (http://www.youtube.com/watch?v=VvqeJkT3uyo)

Jan K
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