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

displaymath55

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

displaymath61

3
Prove that

displaymath63

4
Prove that

displaymath65

5
Conclude that

displaymath67

6
Prove that

displaymath69

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

displaymath73

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

[Calculus] [Back to More Examples]
[Geometry] [Algebra] [Trigonometry ]
[Differential Equations] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Copyright © 1999-2004 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA