MathDB
Romania District Olympiad 2001 - Grade XII

Source:

March 16, 2011
limitintegrationlogarithmscalculusreal analysisreal analysis unsolved

Problem Statement

a)Prove that ln⁡(1+x)≤x, (∀)x≥0\ln(1+x)\le x,\ (\forall)x\ge 0.
b)Let a>0a>0. Prove that:
lim⁡n→∞n∫01xna+xndx=ln⁡a+1a\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}
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