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41 6 Advanced Calculus, Fifth Editionnot uniform. For 0 I x < 1 the error is precisely xn; for this to be less than E, onemust havelog rX"-.log xAs x approaches 1, the value of n required approaches +m. For x = 1 the erroris always 0. Successive members of the sequence f,(x) are plotted in Fig. 6.10 (inSection 6.14). Since the worst errors occur near x = f 1, one can obtain uniformconvergence by restricting x as in the preceding example: -0.5 5 x 5 0.5.6.13 WEIERSTRASS M-TEST FOR UNIFORM CONVERGENCEThe following test is adequate for determining the uniform convergence of a largenumber of familiar series.THEOREM 30 (Weierstrass M-Test) Let C,"=, u,(x) be a series of functionsall defined for a set E of values of x. If there is a convergent series of constantsCz, M, , such thatlu,(x)l~M, forallxinE,then the series Czl U,(X) converges absolutely for each x in E and is uniformlyconvergent in E.Proof. The Weierstrass M-test is first of all a comparison test. For each fixed x,each term of the series C lu,(x)l is less than or equal to the nth term M, of theconvergent series C M,. Hence by the comparison test (Section 6.6, Theorem 12)the series C u,(x) is absolutely convergent.But the comparison series C M, is the same for all x of the range considered.It is from this fact that the uniform convergence follows. For if R, = S - S,(x) isthe remainder after n terms of the series C u,(x), thenas in Theorem 22 of Section 6.9. In other words, if T, denotes the remainder after nterms of the convergent series C M,:thenT, = Mn+l + Mn+2 + .'-,IRn(x)I I Tn.Since 1 M, is a series of constants, for each given E > 0, N can be found such thatT, < E for n 1 N; for this same N, one hasIR,(x)l I T, < E forn 2 N.Since N does not depend on x, but only on r, the uniform convergence has beenestablished. At the same time it has been shown that T, serves as an upper estimatefor the error committed in using only n terms of the series C u,(x), regardless ofthe x. The error will, of course, vary from one x to another, but the upper estimateis the same for all x.
EXAMPLE 1Ezl 5. Here the ratio test gives " ''' ''*an+lpimlzl l---l = n lim +wt . Chapter 6 Infinite Series 41 73fl1 , *, .- - ,?a .
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ContentsVectors and Matrices1.1 Int
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3.6 Combined Operations 183*3.7 Cur
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*6.24 Principal Value of Improper I
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Case of Two Particles 662Case of N
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Differential Calculusof Functionsof
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Vector Integral CalculusPART I.TWO-
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Chapter 5 Vector Integral Calculus
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Functions of aComplex VariableWe as
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Ordinary DifferentialEquations9.1 D
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