Proving a limit of an integral

Question

I don't know how to approach this problem.

Proof that $\lim_{n\to \infty} (n+1)I_n = \frac 12$

where $I_n = \int_{0}^{1} \frac {x^n}{x+1}dx$

You really should explain what you've tried. Where did the problem come from? What's the context? — zhw., Mar 13 '17 at 20:31

score 1 · Answer 1 · answered Mar 13 '17 at 20:30

1

Hint:

$$(n+1)\int_0^1 \frac{x^n}{1+x}\, dx - \frac{1}{2} = (n+1)\int_0^1 x^n\left ( \frac{1}{1+x} - \frac{1}{2}\right )\,dx.$$

answered Mar 13 '17 at 20:30

zhw.

score 1 · Answer 2 · answered Mar 13 '17 at 20:33

1

Hint: Integrate by parts.

$$\int_0^1 \frac{x^n}{1+x} \, dx = \frac{1}{2(n+1)} + \frac{1}{n+1}\int_0^1 \frac{x^{n+1}}{(1+x)^2} \, dx $$

answered Mar 13 '17 at 20:33

RRL

2 Answers2