Show that the series of $f(n)/n^2$, were $f$ is an injective function, diverges.

Question

Consider an injective function $f: \Bbb N \rightarrow \Bbb N$

Show that the series given by:

$$\sum_{n=1}^{\infty } \frac{f(n)}{n^2}$$

diverges.

I'm really not sure how to use the injectivity hypothesis, so any suggestion or any answer would be well received.

Answer 1

Hint. Determine how $f$ would need to be selected in order to make the $N$th partial sum as small as possible.