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

1

Show that , for every n.

2

Show that for all , we have

3

Prove that

4

Prove that

5

Conclude that

6

Prove that

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

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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