Hint : integrate to get an expression for and then set .
(Note : this is an informal proof!)
take the maclaurin series for ln(1+x) which is: x - (x^2)/2 + (x^2)/3 +...(-1)^(k+1) (x^k)/k.
now plug in x=1 and notice that the right side becomes the alternating harmonic series 1 - 1/2 + 1/3 - 1/4 +... so therefore the sum of the alternating harmonic series is ln(2).