# lect118_25_w13 - Monday March 11 Lecture 25 Expressing a...

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Monday, March 11 ±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. Addition and multiplication of power series. 25.1Theorem±Given two power series 6j = 0 to fcj(x ±a) j and 6j = 0 to fbj(x ±a) j which converge to f(x) and g(x) respectively, on the respective intervals I1and I2. Then the series 6j = 0 to f(cj rbj)(x ±a) j converges to f(x)rg(x) on I1I2. 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.