Advanced Calculus fi..

474 Advanced Calculus, Fifth EditionHence for 0 < Ix 10< n;:~i !4 4 . 4 " sin (2n - 1)x., f(x)= -sinx+-sin3x+...=-~n 3nn=l2n-1 'The figure shows the first three partial sums4 4 4 .S1=-sinx, S2=S1+-sin3x, S3=S2+-sinkn 3n 5nof this series. If the graphs are studied carefully, then it becomes clear that f (x) isbeing approached as limit. For x = 0, each partial sum equals 0, so that the seriesdoes converge to the average value at the jump; there is a similar situation at x = f n.However, it should be noted that the approximation to f (x) by each partial sum ispoorest immediately to the left and right of the jump points.EXAMPLE 2 Let f (x) = in + x for -n 5 x 5 0 and f (x) = in - x for0 i x 5 n. The periodic extension of f (x) is the "triangular wave" of Fig. 7.6. Inthis example the extended function is continuous for all x. One findsa,, = IS0 n -, (;+x)cosnxdx+- o (5- X) cos nx cix1"1 2= [i(l - cosnn) + -(I - cosnn) = -(I - cosnn)n n2 n2 ] n2nfor n = 1,2, . . . . For n = 0 a separate computation is needed:The computation of the b,'s is like that of the an's, and one finds b, = 0 forn=l,2, .... Hence4 4 4 " cos (2n - l)xf(x)= -cosx+-cos3x+~~~=-Cn 9n (2n -IT n=1The first two partial sums are plotted in Fig. 7.6. Since there are no jumps, one mustexpect convergence everywhere. It should, however, be noted that at the corners(where f '(x) has a jump), the convergence is poorer than elsewhere.Thus far we have proceeded formally, evaluating coefficients and verifyinggraphically that the series converges to the function. We now proceed to examinethe steps more carefully.The constant term ao/2 of the series is given by the formula