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41 2 Advanced Calculus, Fifth Editionof functions, just as we rounded off the decimal expansion of 4 in Section 6.1. Theprocedure can be justified if the error made is below a prescribed value E that is, if. If (x)- S,(x)l c 6 for a < x < b. However, it can happen, as will be seen below, thatno matter how large n is chosen, the error If (x)- S,(x)l exceeds E for some x ofthe interval a < x < b. This does not contradict the fact that the series converges foreach x; it is simply due to the fact that the number of terms needed to reduce the errorbelow r varies from one x to another in such a manner that no one n will serve forthe whole interval. When this difficulty arises, rounding off over the whole intervala < x fied. On the other hand, it may happen that, however E ischosen, an N can be found such that for each n 2 N, IS,(x) - f (x)l < E for all x ofthe interval a < x c b; if this is so, the series is termed uniformly convergent in theinterval a < x < b. A uniformly convergent series can be rounded off to any accuracydesired:The series x,"=, un(x) is uniformly convergent to S(x) for a set EDEFINITIONof values of x if for each r > 0 an integer N can be found such thatfor n L N and all x in E.The essential idea is that N can be found as soon as E (or the number of decimalplaces desired) is known, without considering which x of the set E is going to beused. The set E must, of course, be a set of values of x for which the series converges,but it need not consist of all such values. In many cases it can be shown that only byrestricting attention to a part of the range for which convergence is known can it beensured that the convergence is uniform. In the worst case, one could restrict attentionto a finite number of values of x, for which the convergence must be uniform.The definition of uniform convergence is given in terms of series; it can equallywell be given in terms of sequences: The sequence f,(x) is uniformly convergent tof (x) for a set E of values of x, if for each E > 0 an N can be found such thatfor n 2 N and all x in E. Thus the uniform convergence of the series C un(x)is equivalent to the uniform convergence of its sequence of partial sums Sn(x).[Actually, sequences and series are simply two different ways of describing thesame limit process; every series can be interpreted in terms of its partial sums, andevery sequence S, can be interpreted as the partial sums of a series, namely, theseries S1 + (SZ - Sl) + . . + (S, - + - - . .]The most common case of uniform convergence is that in which the functionsu,(x) are continuous in a closed interval a ( x 5 b and the series C u,(x)converges in this interval to a continuous function f (x). [It is shown below, inTheorem 3 1 of Section 6.14, that uniform convergence of the series guarantees the
Chapter 6 Infinite Series 41 3Figure 6.8Uniform convergence.continuity off (x).] In such a case the absolute error JR,(x)J in replacement off (x)by the partial sum Sn(x):is continuous for 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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Chapter 2 Differential Calculus of
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PROBLEMSChapter 2 Differential Calc
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Chapter 2 Differential ~alculus of
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magnitudeand the force is inversely
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Vector Integral CalculusPART I.TWO-
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Chapter 5 Vector Integral Calculus
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*5.15 PHYSICAL APPLICATIONSChapter
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Therefore R3 lies in some annulus h
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Fourier Series andOrthogonal Functi
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Functions of aComplex VariableWe as
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Rule IV could also be used, with A
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Ordinary DifferentialEquations9.1 D
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