MathNet 2008-part-05 — 164 problems (EN–VI) Question 6

Let $f(r) = \sum_{j=2}^{2008} \frac{1}{j^{r}} = \frac{1}{2^{r}} + \frac{1}{3^{r}} + \cdots + \frac{1}{2008^{r}}$. Find $\sum_{k=2}^{\infty} f(k)$.