all-quizzes-fa09

all-quizzes-fa09 - CSCE 750, Fall 2009 — Quizzes with...

Info iconThis preview shows pages 1–3. 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: CSCE 750, Fall 2009 — Quizzes with Answers Stephen A. Fenner September 24, 2011 1. Give an exact closed form for ∞ X k =1 2 k 3 k +1 . Simplify your answer as much as possible. Answer: We reduce the expression to a form we’ve already seen in class: ∞ X k =1 2 k 3 k +1 = 2 3 ∞ X k =1 k 3 k = 2 3 ∞ X k =1 kr k , where r = 1 / 3. We saw in class 1 that the sum on the right is ∞ X k =1 kr k = r (1- r ) 2 for all r such that | r | < 1. Thus the final answer is 2 3 · 1 / 3 (1- 1 / 3) 2 = 2 3 · 1 / 3 4 / 9 = 1 2 . 2. Let f a real-valued function defined on R . Recall: • f ( n ) is strictly monotone increasing iff x < y = ⇒ f ( x ) < f ( y ) for all x,y ∈ R . • f ( n ) is strictly monotone decreasing iff x < y = ⇒ f ( x ) > f ( y ) for all x,y ∈ R . 1 If you forgot the formula, re-derive it as follows: (1) start with the formula for an infinite geometric series, ∑ ∞ k =0 r k = 1 / (1- r ); (2) differentiate both sides with respect to r ; (3) multiply both sides by r . 1 Show that if f ( n ) and g ( n ) are both strictly monotone decreasing , then f ( g ( n )) is strictly monotone increasing . Answer: For all x,y ∈ R , we have x < y = ⇒ g ( x ) > g ( y ) ( g is strictly decreasing) = ⇒ f ( g ( x )) < f ( g ( y )) . ( f is strictly decreasing) Thus f ( g ( n )) is strictly increasing. 3. Use the substitution method to show that if T ( n ) = 4 T j n 2 k + n 2 , then T = O ( n 2 lg n ) . Only show the inductive step. Don’t worry about any base case(s)....
View Full Document

This note was uploaded on 12/13/2011 for the course CSCE 750 taught by Professor Fenner during the Fall '11 term at South Carolina.

Page1 / 6

all-quizzes-fa09 - CSCE 750, Fall 2009 — Quizzes with...

This preview shows document pages 1 - 3. Sign up to view the full document.

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