Find the Limit $\lim_{n \to \infty}\sum_{k=1}^{\infty}\frac{k^{n}}{1+k^{n+2}}$ Find $\lim_{n \to \infty}\sum_{k=1}^{\infty}\frac{k^{n}}{1+k^{n+2}}$
My ideas: let $ n \in \mathbb N$ be constant, looking at $\frac{k^{n}}{1+k^{n+2}}$, we know
$$\frac{k^{n}}{1+k^{n+2}}\leq\frac{k^{n}}{k^{n+2}}=\frac{1}{k^{2}}$$
but this does not help me because $\sum_{k=1}^{\infty}\frac{1}{k^{2}}=\pi^{2}/6$
any ideas?

The series equals
$$\frac{1}{2}+ \sum_{k=2}^{\infty}\frac{1}{k^{-n} + k^2}.$$
For each $k\ge 2,$ the terms in the last series increase to $1/k^2$ as $n\to \infty.$ By the monotone convergence theorem, the desired limit is
$$\frac{1}{2} +\sum_{k=2}^{\infty}\frac{1}{k^2} = \frac{1}{2}+\left (\frac{\pi^2}{6} - 1\right) = \frac{\pi^2}{6} - \frac{1}{2}.$$
