This section introduces the use of infinite series to

Chapter , Problem 27

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This section introduces the use of infinite series to solve differential equations. Conversely, differential equations can sometimes be used to sum infinite series. For example, consider the infinite series 1 C 1 1 1 2 C 1 3 C 1 4 1 5 C I note the CCCCpattern of signs superimposed on the terms of the series for the number e. We could evaluate this series if we could obtain a formula for the function f .x/ D 1 C x 1 2x2 C 1 3x3 C 1 4x4 1 5x5 C; because the sum of the numerical series in question is simply f .1/. (a) Its possible to show that the power series given here converges for all x and that termwise differentiation is valid. Given these facts, show that f .x/ satisfies the initial value problem y.3/ D yI y.0/ D y0 .0/ D 1; y00.0/ D 1: (b) Solve this initial value problem to show that f .x/ D 1 3 ex C 2 3 ex=2 cos p3 2 x C p3 sin p3 2 x ! : For a suggestion, see of Section 3.3. (c) Evaluate f .1/ to find the sum of the numerical series given here. 8.2 Series Solu