[SOLVED] sums exist or do not exist

Hi,can some one please help me with the following sums as i need to prove that they either exist or do not exist:

We do NOT say the sum exists if it is infinity.

1. $\displaystyle \sum _{k=1}^{\infty }{{k}^{-1}-{k}^{-2}}$

**I know that: **

**this sequnce $\displaystyle \sum _{k=1}^{\infty }{k}^{-1}$ doesnot exist **

**but this sequence $\displaystyle \sum _{k=1}^{\infty }-{k}^{-2}$does exist**

**so i am confused about if this sequence $\displaystyle \sum _{k=1}^{\infty }{{k}^{-1}-{k}^{-2}}$ exists**

2. i am not sure if the following sum exists

$\displaystyle \sum _{k=1}^{\infty }{{k}^{-1}- \left( k+1 \right) ^{-1}}$