An elementary course of infinitesimal calculus . ng from 0 to any fixed value shortof o. As a matter of fact it will be uniformly convergent up to a;=a,inclusively, but this cannot be established by the above method. + By this notation it is meant that x may range from a to 6 inclusively. L. 35 546 INFINITESIMAL CALCULUS. [CH. Xlll The question may be illustrated graphically* by drawing thecurve y = S (x), and also the approximation-curves y = S„ (a;),for ?i= 1, 2, 3,...t. If the series be uniformly convergent, then,however small the value of o-, from some finite value of n onwardsthe approxim
An elementary course of infinitesimal calculus . ng from 0 to any fixed value shortof o. As a matter of fact it will be uniformly convergent up to a;=a,inclusively, but this cannot be established by the above method. + By this notation it is meant that x may range from a to 6 inclusively. L. 35 546 INFINITESIMAL CALCULUS. [CH. Xlll The question may be illustrated graphically* by drawing thecurve y = S (x), and also the approximation-curves y = S„ (a;),for ?i= 1, 2, 3,...t. If the series be uniformly convergent, then,however small the value of o-, from some finite value of n onwardsthe approximation-curves will all lie within the strip of theplane xy bounded by the cui-ves y = S(x)±(r (18). In the above Ex. the approximation-curve y = Sn (x) is derivedfrom Fig. 17, p. 31, by contracting all the abscissae in the ratio1/ji The curves tend ultimately to lie between the limitsy = +<T for the greater part of their course, but (if <7<1) theywill always transgress these boundaries in the neighbourhood ofthe origin. See Fig. Ex. have If S{x) Fig. 149. -S^ (9;) = tanh nx (19), = lim tanh nx=-l,0, + \ (20), according as x is negative, zero, or positive; see Fig. 22, p. annexed Fig; 150 shews that no approximation-curve lies p f w^^Tj*«*°^^ indebted, here and iu Arts. 196, 198, to a paper byProf. W. F. Osgood, A Geometrical Method for the Treatment of !^ ^^ Certain Double Limits, Bull. Amer. Math. Soc, 2nd ser.,t. o (1896). t See for example Fig. 152, Art. 203. 194-195] INFINITE SERIES. 547 wholly within the required limits, in the immediate neighbour-hood of the origin. We have here an instance of a series whose sum is a dis-continuous function of x, although the individual terms arethemselves continuous. ^ y^ / / «/=5 -A_ 0 Pig. 150. 195. Continuity of the Sum of a Power-Series. We can now shew that the sum of a convergent power-series is a continuous function of x for all values of x withinthe range, of convergence. 35—2 548 INF
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