Integrating Powers and Product of Sines and Cosines: Challenging Problems

The purpose of the following questions is to develop Wallis's formula which has many applications. In particular, for the proof of the Stirling's Formula. For n=0,1,2.., define

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1
Show that tex2html_wrap_inline57 , for every n.
2
Show that for all tex2html_wrap_inline59 , we have

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3
Prove that

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4
Prove that

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5
Conclude that

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6
Prove that

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The Wallis's formula gives tex2html_wrap_inline71 as an infinite product. Indeed, from the previous questions we get

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Note also that the above product can be expressed using factorials. Try to come up with the formula translating the above limit using factorials.

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Author: Mohamed Amine Khamsi

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