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EVALUATION OF FUNCTIONS WHICH FOR CERTAIN VALUES OF
THE VARIABLE BECOME INDETERMINATE.
If u be a function of x of the form , and if for the value x = a, P and Q both vanish ; u, taking the form
, is indeterminate and its true value will be found by differentiating the numerator and denominator separately and taking the quotient of these differentials: that is, using Lagrange's notation, the real value of u will be
But if the same value (ix = a) which makes P and Q vanish also make P = 0, and Q = 0, we must differentiate again, and so on in succession, as long as the numerator and the denominator both vanish when x is put equal to a. Therefore we may say generally that the true value of u when x = a is
p«n) and Q") being the first differential coefficients of P and Q which do not vanish simultaneously when a is put equal to a.
This theory of the evaluation of indeterminate functions was first given by John Bernoulli, Acta Eruditorum, 1704, p. 375.
P(r) The expression in, that is, any one of the series of fractions which present themselves in the operation above described, may be replaced by any equivalent fraction re
sulting from the multiplication or division of both its numerator and denominator by any function of x: the result of the evaluation of this new fraction coinciding with the required value of for the assigned value of x. We may likewise substitute for any finite factor of all the terms of the numerator or denominator of any of the series of fractions, the value which it has when x is put equal to a. These considerations frequently lead to simplifications of the process of evaluation.
This was one of the first functions the value of which was determined in this manner.
(a* + a x + x?)}-(a' - a X + ?)? U =
(a + )! - (a – X)
When x = 0, U = al. One of the most important applications of this process is to find the sums of series for particular values of the variable. The first example was an instance of this, and we shall here add others taken like that from Euler's Calc. Diff. p. 746. (10) The sum of the series
x + 2x + 3 Q3 + ...... + na",
(1 - 2)
2 and we are required to find its value when x = 1, or when the series becomes that of the first order of figurate numbers. By two differentiations we find
O n (n + 1)
(11) The sum of the series
iX + 47+ + 9.73 + ...... + 1".c", 2 + xi? – (n + 1)? 7.+1+ (21+ 2n – 1).2"+? - n":2"+3 is u = --
(1 – x) and we are required to find the real value of u when x = 1, which in this case is the sum of the squares of the natural numbers. Differentiating three times we have
O n (n + 1) (2n + 1)
x - xm+n (12) Let
1- xp i find its value when x = 1. Since u may be put under the form
2." 1 - a" u =
1 + QP 1 - ap!
of which the latter factor alone becomes indeterminate, we need only differentiate that factor so as to find its real value when a = 1, and then multiply it by the value of the first factor when x = 1. The real value of the fraction is