This section introduces the use of infinite series to solve differential equations. Conver...

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

because the sum of the numerical series in question is simply f(1). (a) It’s 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) = y; y(0) = y′(0) = 1, y″(0) = −1.

(b) Solve this initial value problem to show that

For a suggestion, see Problem of Section. (c) Evaluate f(1) to find the sum of the numerical series given here.