FA07Ma1aSol7

FA07Ma1aSol7 - Math 1a Fall Term 2006 SOLUTIONS TO HOMEWORK...

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Unformatted text preview: Math 1a Fall Term 2006 SOLUTIONS TO HOMEWORK 7 Problem 1. The purpose of this problem is to provide an alternative approach to the definitions of log( x ) and e x used in class. (a) Prove that the radius of convergence of the power series X n =0 x n n ! is infinite. Define the limit to be e x . Proof. To find the radius of convergence of the power series X n =0 x n n ! , we apply the ratio test. If x 6 = 0, the ratio of consecutive terms has absolute value x n +1 n ! ( n + 1)! x n = | x | n + 1 . Since this ratio tends towards 0 as n , we conclude that the series converges absolutely for all real numbers x 6 = 0. It also converges for x = 0 by inspection, so it in fact converges for all x R . In other words, its radius of convergence is infinite. (b) Define P n ( x ) = n X j =0 x j j ! , E n ( x ) = X j = n +1 x j j ! the approximation and error. Prove that if | x | < n , then | E n ( x ) | | x | n +1 ( n + 1)! 1- | x | n- 1 . Proof. By definition of E n ( x ) followed by the triangle inequality, | E n ( x ) | = X j = n +1 x j j ! X j = n +1 | x | j j ! . Further simplifying, we see X j = n +1 | x | j j ! = | x | n +1 ( n + 1)! X j = n +1 | x | j- n- 1 j ! / ( n + 1)! | x | n +1 ( n + 1)! X j =0 | x | j n j . Since | x | < n , we have X j =0 | x | j n j = 1- | x | n- 1 . Therefore, | E n ( x ) | | x | n +1 ( n + 1)! 1- | x | n- 1 . 1 2 SOLUTIONS TO HOMEWORK 7 (c) Use the binomial theorem to prove that | P 2 n ( x + y )- P 2 n ( x ) P 2 n ( y ) | e | x | E n ( | y | ) + e | y | E n ( | x | ) . Proof. By definition of P n ( x ), | P 2 n ( x + y )- P 2 n ( x ) P 2 n ( y ) | = 2 n X j =0 ( x + y ) j j !- 2 n X i =0 x i i ! ! 2 n X j =0 y j j ! . Applying the binomial theorem to the first summation, we have | P 2 n ( x + y )- P 2 n ( x ) P 2 n ( y ) | = 2 n X j =0 1 j ! j X i =0 j i x i y j- i- 2 n X i =0 x i i ! ! 2 n X j =0 y j j ! = 2 n X j =0 j X i =0 x i y j- i i !( j- i )!...
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This note was uploaded on 02/22/2010 for the course MA 1a taught by Professor Borodin,a during the Fall '08 term at Caltech.

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FA07Ma1aSol7 - Math 1a Fall Term 2006 SOLUTIONS TO HOMEWORK...

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