_n_k=1^n(n^2)/((n^2+k^2)(n+k)) is equal to:

Question

$\displaystyle\lim_{n	o\infty}\sum_{k=1}^n\frac{n^2}{(n^2+k^2)(n+k)}$ is equal to:

Accepted Answer

This is a Riemann sum: $\displaystyle\lim_{n\to\infty}\frac1n\sum_{k=1}^n\frac1{\left(1+\frac{k^2}{n^2}\right)\left(1+\frac kn\right)} = \int_0^1\frac{dx}{(1+x^2)(1+x)}$

Partial fractions: $\dfrac1{(1+x^2)(1+x)} = \dfrac{1/2}{1+x}+\dfrac{-x/2+1/2}{1+x^2}$

$$\int_0^1\left[\frac1{2(1+x)}-\frac{x}{2(1+x^2)}+\frac1{2(1+x^2)}\right]dx = \frac12\ln2-\frac14\ln2+\frac12\cdot\frac\pi4 = \frac14\ln2+\frac\pi8$$

Solutions · 1