Q:

Compute the $\lim_{n\to\infty}\int_0^1\frac{\mathrm{d}x}{1+x+\frac{x^n}n}$

I know you should prove $f_n (x) = \frac1{1+x+\frac{x^n}n}$ converges uniformly and then the limit follows immediately however I’m struggling to prove the uniform convergence. 

A:

In this case you can even compute it directly:
$\displaystyle 0\le \int_0^1 \frac{1}{1+x}dx-\int_0^1 \frac{1}{1+x+\frac{x^n}{n}}dx\\\displaystyle=\int_0^1 \frac{\overbrace{x^n}^{\le 1}}{\underbrace{(n+nx+x^n)}_{\ge n}\underbrace{(1+x)}_{\ge 1}}dx\le \int_0^1 \frac{1}{n}dx=\frac 1n\to 0$

