Given $N$ and $a$ positive integers, with $a\ge 2$ is it possible to prove the inequality:
$$\sum\_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}\le\frac{N}{2}$$



Upper bound of a series
Since $\displaystyle \frac{k^a}{(k+1)^a+(k+2)^a}<\frac{k^a}{k^a+k^a}=\frac{1}{2}$ then $\displaystyle\sum\_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}<\sum\_{k=1}^N\frac{1}{2}=\frac{N}{2}$.


