This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Ma 116  Calculus II Hw2 (Solutions) Due: Feb 18, 2011 An important component of these written exercises is the quality of your presentation, including: legibility, organization of the solution, and clearly stated reasoning where appropriate. Points will be deducted for sloppy work or insufficient explanations. 1. The meaning of the decimal representation, 0 .d 1 d 2 d 3 is that, .d 1 d 2 d 3 = d 1 10 + d 2 10 2 + d 3 10 3 + where each d i is from the set { , 1 ,..., 9 } . Show that this series always converges. Solution : We prove this using a comparison argument. For each d k we have d k / 10 k 9 / 10 k , and the series, . 99 9 = 9 10 + 9 10 2 + 9 10 3 + = X k =1 9 10 k is a convergent geometric series (ratio = 0.1). 2. The following power series is called the Bessel function of order 0: J ( x ) = X n =0 ( 1) n x 2 n 2 2 n ( n !) 2 . Show that J ( x ) satisfies the secondorder differential equation x 2 J 00 ( x ) + xJ ( x ) + x 2 J ( x ) = 0 . Solution : x 2 J ( x ) = X n =0 ( 1) n x 2 n +2 2 2 n ( n !) 2 = X k =1 ( 1) k 1 x 2 k 2 2 k 2 (( k 1)!) 2 ( k = n + 1) J ( x ) = X n...
View
Full
Document
 Fall '11
 Duboski
 Calculus

Click to edit the document details