Winter 2009 Quiz 1a

Winter 2009 Quiz 1a - MATH 239 Quiz — Winter 2009 SOLUTIONS No calculators or other aids may be used Total marks 20 1(a Let m and n be positive

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

View Full Document Right Arrow Icon

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

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

Unformatted text preview: MATH 239 Quiz — Winter 2009 SOLUTIONS No calculators or other aids may be used. Total marks: 20 1. (a) Let m and n be positive integers. Find an expression (you may use summation notation) for [ x m ](1 + 2 x ) n (1 + x 2 )- 2 . [5 marks] Solution [ x m ](1 + 2 x ) n (1 + x 2 )- 2 = [ x m ] n X j =0 n j 2 j x j ∞ X k =0 ( k + 1)(- 1) k x 2 k ! = [ x m ] n X j =0 ∞ X k =0 (- 1) k ( k + 1) n j 2 j x j +2 k . Since we get an x m whenever j + 2 k = m , we may sum over all possible k and set j = m- 2 k to get [ x m ] n X j =0 ∞ X k =0 (- 1) k ( k + 1) n j 2 j x j +2 k = b m/ 2 c X k =0 (- 1) k ( k + 1) n m- 2 k 2 m- 2 k . (b) Let S be the set of all subsets of the positive integers { 1 , 2 , 3 , . . . } . Define the weight function ω as follows. For the empty set {} , we let ω ( {} ) = 0 and for any non-empty set σ we let ω ( σ ) equal the largest element of the subset σ . For example, if σ = { 1 , 2 , 4 } , then ω ( σ ) = 4, since 4 is the largest element in...
View Full Document

This note was uploaded on 08/09/2009 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

Page1 / 3

Winter 2009 Quiz 1a - MATH 239 Quiz — Winter 2009 SOLUTIONS No calculators or other aids may be used Total marks 20 1(a Let m and n be positive

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