Limit of integral of $\frac{2nx^{n-1}}{1+x},\, x\in [0,1]$

Question

For $n = 1, 2, ...., $ let $$f_n(x)= \frac{2nx^{n-1}}{1+x},\, x\in [0,1] .$$

Find $$ \lim_{n \to \infty} \int \limits_{0}^{1} f_n(x) dx ?$$

The sequence is not uiform convergent on $[0,1]$, because limit does not exist at $1$. How to approach further for the problem. Is dominated convergence theorem applicable?

By integration by parts, you have to study $1+2\int_0^1\frac{x^n}{(1+x)^2}dx$ — Kelenner, Feb 20 '20 at 13:24

score 1 · Accepted Answer · answered Feb 20 '20 at 13:25

1

Applying integration by parts with $u=\frac{1}{1+x}$ and $dv=nx^{n-1}$ yields $$2\frac{x^n}{1+x}|^1_0+2\int \frac{x^n}{(1+x)^2}dx$$ The limit of the first part is clear, and the second part is much more well-behaved.

answered Feb 20 '20 at 13:25

Paul

8,153

Thank you for the answer. – user388189 Feb 20 '20 at 14:16

Limit of integral of $\frac{2nx^{n-1}}{1+x},\, x\in [0,1]$

1 Answers1