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## SEEMOUS 2012 Problem 4

a) Compute $\displaystyle \lim_{n \to \infty} n \int_0^1 \left(\frac{1-x}{1+x} \right)^n dx$.

b) Let $k \geq 1$ be an integer. Compute

$\displaystyle\lim_{n \to \infty}n^{k+1}\int_0^1 \left( \frac{1-x}{1+x}\right)^n x^k dx$.

a) Make the change of variable $y=\frac{1-x}{1+x}$. The limit becomes

$\displaystyle \lim_{n \to \infty} n \int_0^1 y^n \frac{2}{(1+y)^2}dy$

Prove first that if $f$ is continuous then $\displaystyle \lim_{n \to \infty} n \int_0^1 y^n f(y)dy =f(1)$. This can be done in several steps as follows:

• Prove it first for functions of the form $f(y)=y^p,\ p \in \Bbb{Z}_+$.
• Extend it easily to polynomials.
• Approximate $f$ uniformly by polynomials to get the desired result.

For b) something similar works, but it is more intricate. I will come back with the details.