`A = 1 - 1/2 + 1/3 - 1/4 + ... + 1/2011 - 1/2012`

`A = (1 + 1/2 + 1/3 + 1/4 + ... + 1/2012) - 2(1/2 + 1/4 + ... + 1/2012)`

`A = (1 + 1/2 + 1/3 + 1/4 + ... + 1/2012) - (1 + 1/2 + ... + 1/1006)`

`A = 1/1007 + 1/1008 + ... + 1/2012`

`B=1/1007+1/1008+...+1/2012`

`=> A/B`  $\dfrac{  \dfrac{1}{1007} + \dfrac{1}{1008} + ... + \dfrac{1}{2012} }{\dfrac{1}{1007} + \dfrac{1}{1008} + ... + \dfrac{1}{2012} }$ `= 1`

`=> (A/B)^2013 = 1^2023 = 1`

$#ptjuoy$