{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

project1 - (2 k-3 I 2 k-2 ² =-π 4 n X k =1 1 k 2 And so...

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

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!! (

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

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

{[ snackBarMessage ]}

Page1 / 2

project1 - (2 k-3 I 2 k-2 ² =-π 4 n X k =1 1 k 2 And so...

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

View Full Document
Ask a homework question - tutors are online