final07fall

# final07fall - 1 Final Math 262 Fall 2007 Dec. 13 th 9:00 to...

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Unformatted text preview: 1 Final Math 262 Fall 2007 Dec. 13 th 9:00 to 12:00 1. (10 marks) Find the interval of convergence of the power series including the end- points: 1 1 ( 1) 2 ( 3) n n n n x n- =-- Use Ratio Test: 1 1 1 | | 2 | 3| lim lim . lim 2| 3| | | 1 2 | 3| 1 2| 3| 1 . . | 3| 1/ 2 2.5 3.5 n n n n n n n n n a x n n x a n x n x i e x or x for convergence + + + - = =- +- + =- &lt;- &lt; &lt; &lt; For end points x = 2.5 we have 1 2 1 1 1 1 ( 1) 2 1 ( 1) 2 1 . . 1 2 2 n n n n n n diverges I T or p test p n n n-- --- = = -- = For end points x = 3.5 we have 1 1 1 1 ( 1) 2 1 1 ( 1) 2 n n n n n n- -- =- . This is an alt. Series a n = 1/n (b) a n &gt; a n+1 (c) 1 lim 0. n n = (a) Hence interval of convergence is 2.5 3.5 x &lt; 2 Final Math 262 Fall 2007 Dec. 13 th 9:00 to 12:00 2(a) (6 marks) Find the Maclaurin series for the function 1 ( ) . x t e E x dt t- = How many terms of this series are needed to compute E(1) with error less than 0.001? (b) (6 marks) Find the sum of the series 1 1 . n n ne = (a) 2 3 2 ... 1 2! 3! 1 ... 2! 3! t t t t e t t t t + + +- = = + + + and E(x) = 2 2 3 1 1 1 (1 ...) ...; (1) 1 ... 2! 3! 2.2! 3.3! 2.2! 3.3! . ! x t t x x dt x E n n + + + = + + + = + + + . This is not an alt. Series. Error after n-terms = 1 1 1 1 1 ... ... ( 1).( 1)! ( 2).( 2)! ( 1)! ( 2)! ( 3)! n n n n n n n + + &lt; + + + + + + + + + + &lt; 2 1 1 1 ... ( 1)! ( 2) ( 2) n n n + + + + + Expression in square bracket is a Geo. Series = 3 1 1 2 10 1 ( 1)! ( 1) ( 1)! 1 2 n n n n n- + = &lt; + + + - + . n = 7 accepted (b) 2 3 1 1 1 ... 2 3 n n e e e ne-- - = = + + + Note 2 3 ln|1 | ... 2 3 x x x x-- = + + + |x | &lt; 1 i.e....
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## This note was uploaded on 01/15/2010 for the course MATH MATH 262 taught by Professor Gregrix during the Spring '09 term at McGill.

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final07fall - 1 Final Math 262 Fall 2007 Dec. 13 th 9:00 to...

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