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#1 2011-06-17 05:28:34

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,612

Grandi's series (1-1+1-1...)

Grandi's series is the infinite sum: 1-1+1-1...

What's interesting about it is that it cna have several 'values':
1. We can group the terms like this:
(1-1)+(1-1)...=0+0+0+0+0+...=0
2. We can group the terms this way:
1+(-1+1)+(-1+1)...=1+0+0+...=1
3. The final way that I know of is to have s be equal to the sum:
s=1-1+1-1...
s=1+(-1+1-1...)
s=1-(1-1+1-1...)
s=1-s
2s=1
s=1/2
which is quite an unusual answer.

http://en.wikipedia.org/wiki/Grandi's_series

Last edited by anonimnystefy (2011-06-17 05:28:48)


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#2 2011-06-17 16:25:36

soroban
Member
Registered: 2007-03-09
Posts: 452

Re: Grandi's series (1-1+1-1...)


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#3 2011-07-09 16:58:30

MathPro101
Member
Registered: 2011-06-08
Posts: 18

Re: Grandi's series (1-1+1-1...)

Question: When I explained Grandi's series to a friend, they stated that addition is commutative, which is true. They wanted to know why the commutative property is allowed to be violated for this series. I suggested that it has to do with the fact that it is a series, and not just any normal addition. Could anyone help me explain why we don't break any laws? Thank you.

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