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

Let $f$ be any given bijection from $\Bbb N\to\Bbb N$. Consider the partial sum $S_n=\sum_{k=1}^n\frac{f(k)}{k^2}$. Suppose $\{t_i\}_{i=1}^n$ is a strictly increasing sequence such that $t_i\in f(\{1,2,...,n\})$. Clearly, $t_i\ge i$. Then, from the rearrangement inequality$$S_n\ge\sum_{i=1}^n\frac{t_i}{i^2}\ge\sum_{i=1}^n\frac i{i^2}$$

Thus, $\lim_{n\to\infty}S_n\ge\lim_{n\to\infty}\sum_{i=1}^n\frac i{i^2}\to\infty$.