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everybody knows that

[tex]log(1+x) = x-\frac{x^2}{2} + \frac{x^3}{3} - ...[/tex]

plug in x=1 & the series converges & we get

log2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - ...

2log2 = 2 - 1 + 2/3 - 1/2 + 2/5 - 1/3 + 2/7 - ...

take the terms together which have a common denominator (ie simplify) & we get

2log2 = 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 - ... = 1 - 1/2 + 1/3 -1/4 + ... = log2

hence 1 = 2

QED

here's a similar one

log2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - ...

= (1 + 1/3 + 1/5 + 1/7 + ...) - (1/2 + 1/4 + 1/6 + 1/8 + ...)

= {(1 + 1/3 + 1/5 + ...) + (1/2 + 1/4 + 1/6 + ...)} - 2(1/2 + 1/4 + 1/6 + ...)

= (1 + 1/2 + 1/3 + ...) - (1 + 1/2 + 1/3 + ...)

= 0

i guess the problem must have something to do with the 'simplification' & doing something to an infinite sum. off the top of my head those are my guesses