project1 - (2 k-3)!! I 2 k-2 =- 4 n X k =1 1 k 2 And so we...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Take-Home Quiz Due March 19 Directions: This project (take-home quiz) is designed to be done in a group of 3 people. In your write-up, please state how each person contributated. Only turn in ONE solution per group. Each group will receive one grade. Please double check your partners work since it will affect your grade. This project is to prove that n =1 1 /n 2 = π 2 / 6. There are a lot of different proofs out there, but this one uses the techniques from this class. For all integers n , define: I 2 n = Z π/ 2 0 t 2 cos 2 n ( t ) dt 1. By applying integration by parts twice show that Z π/ 2 0 cos 2 n ( t ) dt = - 2 n 2 I 2 n + n (2 n - 1) I 2 n - 2 2. Show Z π/ 2 0 cos 2 n ( t ) dt = (2 n - 1)!! (2 n )!! · π 2 Where we use the notation !! to mean: (2 n )!! = 2 · 4 · 6 ··· (2 n - 2)(2 n ) (2 n + 1)!! = 1 · 3 · 5 ··· (2 n - 1)(2 n + 1) and 0!! = 1 and (-1)!!=1 (HINT: See problem 7.1.43-46 for an idea) Thus, we have - 2 n 2 I 2 n + n (2 n - 1) I 2 n - 2 = (2 n - 1)!! (2 n )!! · π 2 3. Show that (2 n )!! (2 n - 1)!! I 2 n - (2 n - 2)!! (2 n - 3)!! I 2 n - 2 = - π 4 n 2 4. Deduce that we then get (2 n )!! (2 n - 1)!! I 2 n - 0!! ( - 1)!! I 0 = n X k =1 ± (2 k )!! (2 k - 1)!! I 2 k - (2 k - 2)!!
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (2 k-3)!! I 2 k-2 =- 4 n X k =1 1 k 2 And so we have (2 n )!! (2 n-1)!! I 2 n = 3 24- 4 n X k =1 1 k 2 = 4 2 6-n X k =1 1 k 2 5. State why it is sucient to show that lim n (2 n )!! (2 n-1)!! I 2 n = 0 in order to conclude that n =1 1 /n 2 = 2 / 6. 6. Verify that I 2 n = Z / 2 t 2 cos 2 n ( t ) dt 2 2 Z / 2 sin 2 ( t ) cos 2 n ( t ) dt 2 2 Z / 2 sin 2 ( t ) cos 2 n ( t ) dt = 2 2 Z / 2 cos 2 n ( t ) dt-Z / 2 cos 2 n +2 ( t ) dt 7. Show that 2 2 Z / 2 cos 2 n ( t ) dt-Z / 2 cos 2 n +2 ( t ) dt = 3 8 (2 n-1)!! (2 n )!!-(2 n + 1)!! (2 n + 2)!! = 3 8 (2 n-1)!! (2 n + 2)!! 8. Deduce that < (2 n )!! (2 n-1)!! I 2 n 3 8 1 2 n + 2 9. Why are we done?...
View Full Document

This note was uploaded on 07/11/2011 for the course MAC 2312 taught by Professor Bonner during the Spring '08 term at University of Florida.

Page1 / 2

project1 - (2 k-3)!! I 2 k-2 =- 4 n X k =1 1 k 2 And so we...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online