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

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