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Q&A #267

Teachers' Lounge Discussion: An algebraic "proof" that 2 = 1

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From: Mr. Leohr <jnmleohr@adelphia.net>
To: Teacher2Teacher Public Discussion
Date: 2003111900:50:18
Subject: Another fun proof that shows 2 = 1

Here's another proof that shows 2 = 1 that requires some knowledge of
infinite series.

If   S = 1 - 1/2 + 1/3 - 1/4 + 1/5 ...
and 2S = 2 - 1   + 2/3 - 1/2 + 2/5 - 1/3 + 2/7 - 1/4 ...
which can be simplified by grouping terms to
    2S = (2-1) - (1/2) + (2/3 - 1/3) - 1/4 ...
    2S = 1 - 1/2 + 1/3 - 1/4 + 1/5 ...

So 2S = S and if we divide both sides by S then

2 = 1

Here's where we went wrong...

In infinite series, you are not allowed to reorder the elements.  It
is not commutative under addition.  Crazy huh?

Mr. Leohr

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