Why does the series $\sum\limits_{k=1}^∞ \frac{k}{k^2 + 1} $ diverge?

Question

Why does $\sum\limits_{k=1}^\infty \Re\mathopen{} \left(\frac{k-i}{k^2 + 1}\right)\mathclose{}=\sum\limits_{k=1}^\infty \frac{k}{k^2 + 1} $ diverge?

How should I go about approaching this problem?

Answer 1

We have

$$\text{Re}\mathopen{}\left(\frac{k+i}{k^2+1}\right)\mathclose{}=\frac{k}{k^2+1}\ge\frac1{2k}$$

and the harmonic series diverges.

Answer 2

$\begin{array}\\ \sum\limits_{k=1}^n \Re \left(\frac{k-i}{k^2 + 1}\right) &=\sum\limits_{k=1}^n \Re \left(\frac{k}{k^2 + 1}\right) +\sum\limits_{k=1}^n \Re \left(\frac{-i}{k^2 + 1}\right)\\ &=\sum\limits_{k=1}^n \frac{k}{k^2 + 1}\\ \text{so}\\ \big|\sum\limits_{k=1}^n \Re \left(\frac{k-i}{k^2 + 1}\right)\big| &=\big|\sum\limits_{k=1}^n \frac{k}{k^2 + 1}\big|\\ &>\big|\sum\limits_{k=1}^n \frac{k}{k^2}\big|\\ &>\big|\sum\limits_{k=1}^n \frac{1}{k}\big|\\ \end{array} $

and this last sum diverges.