Grandi’s Series

If you look at the series:

S = 1 – 1 + 1 -1 + 1 -1 …to infinity
the sum turns out to be , S=0.

Now , if we look it in another way

S = 1 -(1-1) – (1-1) – (1-1)…to infinity
the Sum turns out to be , S=1

The result is either 0 or 1.

The most interesting part comes now.
A priest and Mathematician named Grandi, presented an alternate but mind boggling proof for the summation.

Let us say ,
S = 1-1+1-1+1-1…

Now say,
S = 1-(1-1+1-1+1 …)

So that makes it,
2S = 1
or
S =1/2

The most fun part is the result , sum of all the integers being a fraction (1/2).

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