By V. I. Smirnov and A. J. Lohwater (Auth.)
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Extra info for A Course of Higher Mathematics. Volume I
As before, we shall take along with x its corresponding point K, displaced on the axis OX. Let the point K move in such a way t h a t , however large an interval T'T we take, with the origin as centre, t h e point K will eventually be displaced outside it, and from then on will remain outside. I n this case, x is an infinitely large magnitude, and tends to infinity. Let 2M be the length of the interval T'T. Recalling 29] INFINITELY LARGE MAGNITUDES 55 that the length of the interval OK = \x\, we can give the following definition: The magnitude x is said to be infinitely large, or to tend to infinity, if on successive variation of x, \x \ becomes, and on further variation remains, greater than any given positive number M.
2> · · · S k· Suppose that 6 > 0 and q < 0. The difference 6/(1 — q) — sn is now positive for even n and negative for odd n, so that the variable sn is alternately greater than, and less than, the limit to which it tends. The same remarks apply in the case of magnitudes that tend to a given limit as were made in the previous paragraph, apropos magnitudes that tend to zero. Any constant, equal to the number a, comes under the definition of a variable, tending to the limit a. We note here, that a magnitude, all of whose values are equal to a, has in the ordinary way an infinite set of values, though all these values are equal to the same number.
29. Infinitely large magnitudes· If the variable x tends to a limit, it is evidently bounded, as already remarked. We now consider some cases of variation of unbounded magnitudes. As before, we shall take along with x its corresponding point K, displaced on the axis OX. Let the point K move in such a way t h a t , however large an interval T'T we take, with the origin as centre, t h e point K will eventually be displaced outside it, and from then on will remain outside. I n this case, x is an infinitely large magnitude, and tends to infinity.
A Course of Higher Mathematics. Volume I by V. I. Smirnov and A. J. Lohwater (Auth.)