Home > Analysis, Olympiad, Problem Solving > SEEMOUS 2012 Problem 4

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.

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