u = x" log a when x = 0, n being positive.

0, when a = ∞.

(34)

Differentiating three times, we find the real value to be

2 =

which is the sum of the reciprocals of the squares of the natural

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.

Functions which for a particular value of the variable take the form 0, 0, 1, may be reduced to a shape in which the preceding methods are applicable. Let x and y be functions of a and u = , then if for x = a

reduced to the determination of y log ≈, which takes the form

(42) Find the value of when a = 0, a being positive. log, and we have to find the value of

This is equivalent to

x log a when x = 0.

Now by Ex. (34) xa log x = 0 when x =

= 0.

Therefore

As before, a log x = 0 when ≈ = 0;

therefore u = 6° = 1 when a = 0.

We may arrive at the value of this function

consideration that, when a is indefinitely diminished,

by the sin x

1,

or sin x = x therefore when a = 0, xsin2 = x2 = 1, by Ex. (42).

In the same way it would appear that

0

(45) u = (cot a) since when a = 0.

By similar considerations it appears that

\ (46)

(cot a) sin r = 1 when a = 0.

u = (1 + nx) when ≈ = 0.

u = (1 +nx)2 = e” when a = 0.

This result may be verified by expanding u by the binomial theorem: that gives

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 ƒ (x) which shall be equal to p(x) from x = ∞ to an, and to (x) from an to a∞, he writes

f (2) = (1 - 0) ¢ (b) + 0-n y (2).

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.