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# An Associative fallacy

## Question

Consider the following sequence where

(1 + -1) + (1 + -1) + (1 + -1) + (1 + -1) + (1 + -1) + (1 + -1) ... now clearly this is the same as (0) + (0) + (0) + (0) + (0) + (0) ... = 0

however if I apply the associative property of addition to this series I get...

1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1)  ... which clearly equals 1 + (0) + (0) + (0) + (0) + (0) ... = 1

But 1 does not 0, is the associative property wrong?

## Recommended Posts

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Association holds for finite series. Infinite series converge only under certain conditions, which fail in this case. So it's not proper to give the series a value at all. But let's ignore that and call the series S anyway. Then what is S? Clearly it's 1/2:

Spoiler

S = 1-1+1-1+1-1+1-...
0+ S = 0+1-1+1-1+1-1+1-...
--------------------------
0+2S = 1+0+0+0+0+0+... = 1

2S=1; S=1/2.

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My favorite one:

Take the sum of all the integers. Call it S.
Take the sum of even integers. That sum is S/2.
Because S is infinite, S/2=S, their difference is zero.
So the sum of the odd integers is zero.

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