MathDB
G 16

Source:

May 25, 2007
algebrapolynomialintegrationinequalitiesIrrational numbers

Problem Statement

For each integer n≥1n \ge 1, prove that there is a polynomial Pn(x)P_{n}(x) with rational coefficients such that x4n(1−x)4n=(1+x)2Pn(x)+(−1)n4nx^{4n}(1-x)^{4n}=(1+x)^{2}P_{n}(x)+(-1)^{n}4^{n}. Define the rational number ana_{n} by an=(−1)n−14n−1∫01Pn(x)  dx,  n=1,2,⋯ .a_{n}= \frac{(-1)^{n-1}}{4^{n-1}}\int_{0}^{1}P_{n}(x) \; dx,\; n=1,2, \cdots. Prove that ana_{n} satisfies the inequality ∣π−an∣<145n−1,  n=1,2,⋯ .\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.
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