sorry to bother you like that, I got the solution but missed several parts in your way :

1. for P(k+1) isn't $\displaystyle -\frac{1}{2k}$ missing on the LHS:

$\displaystyle 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....-\frac{1}{2k}-\frac{1}{2(k+1)}=\frac{1}{k+2}+\frac{1}{k+3}+\frac {1}{k+4}+...+\frac{1}{2(k+1)}$

so that later I could use P(k) LHS to RHS?

2. did you only apply actions to P(k) or did I get it wrong?

3. where did that last part come from - $\displaystyle \frac{1}{k+2}+\frac{1}{k+3}+....\frac{1}{2k}+\frac {1}{2k+1}+\left(\frac{1}{k+1}-\frac{1}{2k+2}\right)$?

anywise, thanks a lot!