Advanced Calculus fi..

Chapter 6 Infinite Series 449THEOREM 50 Let f (x) be continuous and let f (x) 2 0 for a 5 x -Z ao. The2the integrallWf(x)dxconverges and has value I if and only if the seriesconverges and has sum I.If f (x) changes sign infinitely often, then convergence of the integral will certainlyimply that of the series; however, the converse need not hold, as the exampleliD sin 2irx dx (6.72)shows (Problem 2 following Section 6.25). The proper connection between seriesand integrals in this case is given by the following theorem.THEOREM 51 Let f (x) be continuous for a 5 x c oo. Let f (x) 2 0 for a =bo 5 x i bl, f (x) 5 0 for bl 5 x 5 b2 and, in general, (-1)" f (x) 2 0 forb, 5 x _( bn+l, where b, is a monotone sequence such thatThen the integrallim b, = oo.n-+mconverges and has value I if and only if the alternating seriesiconverges and has sum I.Proof.If the integral converges to I, then as before,lim lb" f(x)dx = lim(al + +a,) = 1,n-twn+caso that the series converges and has sum I.Conversely, let the series converge to I. Let 6 > 0 be given and choose N so"-large that1 1la~+...+a,-I1