By V. I. Smirnov and A. J. Lohwater (Auth.)

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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. In other words, the magnitude x is called infinitely large if it satisfies the following condition: given any positive number M, there exists a value of x such that, for all subsequent x, \ x | > M.

G > i ) , (β) and let M be any given positive number. e. the variable in question tends to + °°. 56 FUNCTIONAL RELATIONSHIPS AND THE THEORY OF LIMITS [30 If q is replaced b y (-— q) in the sequence (5), the only change is in the signs of odd powers of q, the absolute values of the members of the sequence remaining as before; thus, for negative q, with absolute value greater t h a n unity, the sequence (5) tends to infinity. When in future we say t h a t a variable tends to a limit, a finite limit is to be understood.

Hence, using the second property of infinitesimals [26], the difference &/(l — q) — sn is an infinitesimal, and we can say that the constant 6/(1 — q) is the limit of the sequence ^1? $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.