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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...
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This note was uploaded on 10/09/2011 for the course MATHEMATIC MA116 taught by Professor Duboski during the Fall '11 term at Stevens.
 Fall '11
 Duboski
 Calculus

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