MathDB

2

Part of 2005 Brazil Undergrad MO

Problems(1)

Sequence

Source:

10/24/2005
Let ff and gg be two continuous, distinct functions from [0,1]→(0,+∞)[0,1] \rightarrow (0,+\infty) such that ∫01f(x)dx=∫01g(x)dx\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx Let yn=∫01fn+1(x)gn(x)dxy_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}, for n≥0n\geq 0, natural. Prove that (yn)(y_n) is an increasing and divergent sequence.
functionintegrationinequalitiesinductionreal analysisreal analysis unsolved