real analysis - Chapter 12 Arithmetic of Power Series Introduction In Chapter 7 we have dealt with the representability of a function by a power series

real analysis - Chapter 12 Arithmetic of Power Series...

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Chapter 12 Arithmetic of Power Series Introduction. In Chapter 7, we have dealt with the representability of a function by a power series. A natural question arises as to the representability of the sums, products and quotients of functions by power series in terms of the power series of the functions and whether we can formally add, multiply or divide the power series representing the functions to give a power series for the sum, product and quotient. 12.1 Sums of Power Series The situation of representing sum of functions by power series is quite simple. Theorem 1 . Suppose the function f is represented by with radius of convergence r 1 ± n = 0 a n x n and g is represented by with radius of convergence r 2 . Then the sum f + g is ± n = 0 b n x n represented by the power series with radius of convergence r min( r 1 , r 2 ). ± n = 0 ( a n + b n ) x n Proof. If min( r 1 , r 2 ) = 0, then we have nothing to prove. Suppose now 0 < min( r 1 , r 2 ). Let c be a real number such that 0 < c < min( r 1 , r 2 ). Then the n -th partial sum s n ( x ) = ± k = 0 n a k x k converges absolutely and uniformly to f on [ c , c ] by Theorem 4 and Remark 16 of Chapter 7 since c < r 1 . Similarly we deduce that the n -th partial sum converges absolutely t n ( x ) = ± k = 0 n b k x k and uniformly to g on [ c , c ], since c < r 2 . This means given any ε > 0, there exists a positive integer N 1 such that for all integer n N 1 and for all x in [ c , c ], --------------------------------- (1) | s n ( x ) − f ( x ) | < ² 2 and there exists a positive integer N 2 such that for all integer n N 2 and for all x in [ c , c ], --------------------------------- (2). | t n ( x ) − g ( x ) | < ² 2 Let N = max ( N 1 , N 2 ). It follows by (1) and (2) that for all integer n N and for all x in [ c , c ], . | s n ( x ) + t n ( x ) − ( f ( x ) + g ( x )) | [ | s n ( x ) − f ( x ) | + | t n ( x ) − gf ( x ) | < ² 2 + ² 2 = ² This means that converges uniformly to f + g on [ c , c ]. Take a s n ( x ) + t n ( x ) = ± k = 0 n ( a k + b k ) x k real number c' such that 0 < c < c' < min( r 1 , r 2 ). Then by Theorem 4 of Chapter 7, is convergent. It follows by Proposition 5 Chapter 7 that s n ( c ) + t n ( c ) = ± k = 0 n ( a k + b k )( c ) k converges absolutely to f + g on [ c , c ]. (We may also deduce this s n ( x ) + t n ( x ) = ± k = 0 n ( a k + b k ) x k directly by noting that for | x | < c , and are convergent and since for each ± n = 0 a n x n ± n = 0 b n x n integer n 0, , by the Comparison Test, is ( a n + b n ) x n [ a n x n + b n x n ± n = 0 ( a n + b n ) x n convergent for | x | c . ) Therefore, converges for any x such that | x | c and for ± n = 0 ( a n + b n ) x n © Ng Tze Beng