Having trouble with this infinite series and deciding whether it converges or diverges.
The series:
$$\sum_{n=1}^\infty \frac{1}{2+i^n}$$
I tried using the ratio test but get stuck in the computation. Ended up with something like $$\lim_{n\to \infty}\frac{2+i^n}{2+i^n*i}$$
Do I have to take the modulus of the rational first? what's the modulus of $i^n$?
The absolute value of the term within the summation does not tend to zero, so the summation diverges. The value of $i^n$ is either $i,-1,-i$ or $1$ and hence the value of the summation is: $$\frac{1}{2+i}+\frac{1}{2-1}+\frac{1}{2-i}+\frac{1}{2+1}+...$$ $$=\frac{2-i}{5}+1+\frac{2+i}{5}+\frac{1}{3}+...$$ ...which clearly diverges.