Find $$\lim_{n\to \infty}\int_1^{\infty}\frac{n}{1+x^n}dx$$
Lebesgue dominated convergence does not work. My next thought was to use Fatou's lemma and reverse Fatou's lemma to bound the integral, but I haven't the faintest idea how to calculate the liminf or limsup of the integrand. Wolfram Alpha tells me that the value of the integral decreases (starting with n=2) from about 1.2 to about 0.7, but I can't even show that the sequence is decreasing right now.
Any hints would be appreciated.
Set $u=x^n$ to get: $$I_n:=\int^\infty_1\frac{n}{1+x^n}\,dx=\int^\infty_1 \frac{1}{u^\frac{n-1}{n}(1+u)}\,du$$ For $n\geq 2$, the inequality $u^\frac{n-1}{n}\geq u^\frac{1}{2}$ holds pointwise for $u\in [1,\infty)$ so: \begin{align} \left| \frac{1}{u^\frac{n-1}{n}(1+u)}\right|\leq \frac{1}{\sqrt[]{u}(1+u)} \end{align} Hence by dominated convergence theorem we get: \begin{align} \lim_{n\to\infty}I_n = \int^\infty_1 \lim_{n\to\infty}\frac{1}{u^\frac{n-1}{n}(1+u)}\,du=\int^\infty_1 \frac{1}{u(1+u)}\,du = \log(2) \end{align} And indeed $\log(2)\approx 0.7$.
