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Unformatted text preview: Tuesday, March 6 Lecture 25 : Expressing a power series as a function and operations on power series. (Refers to Section 8.7 ) After having practiced the problems associated to the concepts of this lecture the student should be able to : Determine the sum, the difference and the product of two power series, predict how the derivative and integration of a power series changes the function it converges to, express a power series which is a relative of the geometric series as a function. 25.1 Theorem Given two power series j = 0 to c j ( x a ) j and j = 0 to b j ( x a ) j which converge to f ( x ) and g ( x ) respectively, on the respective intervals I 1 and I 2 . Then the series j = 0 to ( c j b j )( x a ) j converges to f ( x ) g ( x ) on I 1 I 2 . Proof omitted. 25.1.1 Example Suppose the series is known to converge only on the interval [ 3, 3) while the series is known to converge only on the interval ( 1, 1). Determine the terms which describe the sum of the two series and state its interval of convergence. The interval of convergence is the intersection of the two intervals of convergence, i.e., ( 1, 1)....
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