Adding complex numbers in a sum

Question

Basically, I have to deal with this. It seems easy but I have a hard time with it personally.

$$\sum_{n=0}^{\infty}\left(\frac{1}{n+1+\textbf{i}}-\frac{1}{n+\textbf{i}}\right)$$

I tried several different things to write this mess as a single fraction.

Answer 1

It's a telescoping series. So, the sum is$$-\frac1{0+1+i}+\lim_{n\to\infty}\frac1{n+1+i}=-\frac1{1+i}.$$

Answer 2

The sum of the first $n$ terms is $-\frac 1 i +\frac 1 {n+i}$. Hence the sum is $-\frac 1 i$.