# How do i evaluate this sum?

Let $[x]$ be the nearest integer to $x$. (for $x=n+\frac{1}{2}, n \in N$, let $[x]=n$).

Find the value of $$\displaystyle\sum_{m=1}^{\infty} [\sqrt m]^{-3}$$

• Do you have any ideas about what numbers are $[\sqrt m] ^2$? – Karolis Juodelė Jun 28 '14 at 13:40

Hint: note that $[m]=1$ twice, it equals $2$ four times (from $3$ through $6$). How many times does it equal $k$?