aC - (i) k = 1 k 2 k . (ii) k = k ( k + 1 ) 3 k . (iii) k =...

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

View Full Document Right Arrow Icon
University of Toronto, MAT 137Y, 2008–2009 Problem Set Supplement #3 Not to be handed in. This assignment will not count towards your course mark. The following exercises cover new material which you are responsible for the final exam. 1. SHE Section 12.5: 4, 6, 12, 16, 28, 32, 46. 2. SHE Section 12.8: 6, 8, 10, 14, 16, 26, 46. 3. SHE Section 12.9: 2, 6, 14, 28, 42. 4. Find the radius of convergence for the following power series. (i) k = 1 k k k ! x k . (ii) k = 1 k k ( k ! ) 2 x k . (iii) k = 1 k k ( k ! ) 3 / 2 x k . (iv) k = 1 ( k ! ) 1 / k x k . 5. Starting with the formula 1 1 - x = 1 + x + x 2 + ··· , show that x d dx ± x d dx ± 1 1 - x ²² = x + 2 2 x 2 + 3 2 x 2 + ··· , and hence derive a formula for k = 1 k 2 x k . 6. Let { a n } be the Fibonacci sequence defined by a 0 = a 1 = 1 and a n + 2 = a n + 1 + a n for n 0. (a) Prove that a n 2 n . (b) Let f ( x ) = n = 0 a n x n . Show that this power series has radius of convergence R 1 2 . (c) By computing f ( x ) - xf ( x ) - x 2 f ( x ) , show that f ( x ) = 1 / ( 1 - x - x 2 ) . (d) Deduce that a n = f ( n ) ( 0 ) n ! . 7. Evaluate the following series by converting each one to a power series.
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: (i) k = 1 k 2 k . (ii) k = k ( k + 1 ) 3 k . (iii) k = k 2 2 k . (iv) k = 1 2 k k ( k + 1 ) . 8. Find a power series expansion for f ( x ) = Z x e t 2 dt valid for all x . 9. Suppose a = 1, a 1 = 1, and a n + 2 = a n + 1 + 6 a n for n 0. (a) Find a 2 , a 3 , a 4 . (b) Prove that a n 6 n . (c) Let f ( x ) = n = a n x n . Show that this power series has radius of convergence R 1 6 . (d) Show that f ( x ) = 1 / ( 1-x-6 x 2 ) . (Hint: This is similar to a previous exercise.) (e) Use partial fractions to write f ( x ) = A 1-3 x + B 1 + 2 x for some constants A and B . (f) Use the power series expansions of 1 1-3 x and 1 1 + 2 x to show that a n = 3 5 3 n + 2 5 (-2 ) n ....
View Full Document

This note was uploaded on 04/09/2010 for the course MAT 137 taught by Professor Uppal during the Spring '08 term at University of Toronto- Toronto.

Page1 / 2

aC - (i) k = 1 k 2 k . (ii) k = k ( k + 1 ) 3 k . (iii) k =...

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