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(34) U = 2" log x when x = 0, n being positive.
and therefore by the last example
u = 2" log x = 0, when x = 0.
Differentiating three times, we find the real value to be
which is the sum of the reciprocals of the squares of the natural numbers.
(38) The sum of the series
. + &c. to
which is therefore the sum of the squares of the reciprocals of the odd numbers.
The reader will find other examples of a similar kind relating to the summation of series in Euler's Calc. Diff: p. 760, seq.
Sometimes the value of an indeterminate function may be most readily found by throwing it into a form in which its real nature is more easily seen.
(39) If u = 24 sin find its value when x = 0 .
} = 0.00, when
Functions which for a particular value of the variable take the form 0° 0° 1c, may be reduced to a shape in which the preceding methods are applicable. Let x and y be functions of w and u = *", then if for x = a
Now since % = elog?, u = eyloga ; and these three cases are reduced to the determination of y log %, which takes the form 0 x $o.
De Morgan's Diff: Calc. p. 175. (42) Find the value of 2020 when x = 0, a being positive. This is equivalent to fita loga, and we have to find the value of 219 log x when x = 0. Now by Ex. (34) 29 log x = () when x = 0. Therefore
zita = e = 1 when x = 0. If a be negative 2020 = 0) when x = 0). (13) Find the value of
u= C) = 6" when x = 0.
(44) u = pisin x when x = 0.
sin x consideration that, when x is indefinitely diminished, — =1, or sin x = x : therefore when x = 0, æsin x = level = 1, by Ex. (42). In the same way it would appear that
(sin x) sin 2 = 1 when x = 0.
it appears that sin x . log x = 0 when x = 0); and similarly that sin x. log (sin x) = () when x = 0.
(45) u = (cot x)SİN 2 = 60" when x = 0.
(cot x) sin ! = 1 when l= 0).
log (1 +- nr) Here u = e , and we have to find the value of log (1 + nx)
? when w = 0. Differentiating we find this to
This result may be verified by expanding u by the binomial theorem: that gives
1 1 na 202 1 / 1 u=l+
Functions which for a particular value of the variable take the form o', have been used by Libri to introduce discontinuity into ordinary functions.
Thus, if it be desired to express a function f (x) which shall be equal to ♡ (w) from x = -00 to « = n, and to y (x) from X = n to x = 0, he writes
f(x) = (1 - 004") 0 (0) +00** ¥ (w). See his Mémoires de Mathématique et de Physique, Vol. 1. p. 44, and Crelle's Journal, Vol. x. p. 303. In the same Journal, Vol. x11. p. 134 and p. 292, the reader will find some discussion on the real value of this indeterminate expression,