Today's calculation of Integral 557
Source: 2010 Hosei University entrance exam/Bioscience and Applied Chemistry
February 16, 2010
calculusintegrationlimittrigonometryfloor functionfunctionanalytic geometry
Problem Statement
Find the folllowing limit.
\lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).