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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.
 Fall '08
 Borodin,A
 Calculus, Linear Algebra, Algebra, Power Series

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