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Reciprocal of odd double factorial series

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October 31, 2019
factorialseriesDefinite integralreal analysisintegral sequence

Problem Statement

Consider the sequence (In)n≥1, \left( I_n \right)_{n\ge 1} , where In=∫0π/4esin⁡xcos⁡x(cos⁡x−sin⁡x)2n(cos⁡x+sin⁡x)dx, I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, for any natural number n. n.
a) Find a relation between any two consecutive terms of In. I_n.
b) Calculate lim⁡n→∞nIn. \lim_{n\to\infty } nI_n.
c) Show that ∑i=1∞1(2i−1)!!=∫0π/4esin⁡xcos⁡x(cos⁡x+sin⁡x)dx. \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx.
Cătălin Țigăeru
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