Given $N$ and $a$ positive integers, with $a\ge 2$ is it possible to prove the inequality: $$\sum_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}\le\frac{N}{2}$$
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Since $\displaystyle \frac{k^a}{(k+1)^a+(k+2)^a}<\frac{k^a}{k^a+k^a}=\frac{1}{2}$ then $\displaystyle\sum_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}<\sum_{k=1}^N\frac{1}{2}=\frac{N}{2}$. 

