Plane trigonometry, by S.L. Loney.

298 TRIGONOMETRY.
and that
X2
a = 1 +x loge a + (logea)2 +... ad inf:
When n is greater than unity, we have
{( + ) i + ( l)
1 nx (nx -) 1 nx (nx - l) (x - 2) 1
= 1 +nx -+ n...
1.2 n 1. 2.3 n3
- - x x - -)I x - -
=1 +-x —n + + —...
1.2 1.2.3
In this expression make n infinitely great. The lefthand becomes, as in Art. 250, ex.
The right-hand becomes
X2 x3
1 + + +...
Hence we have
X2 X3
ex= 1+x +-x + +...ad inf.......(1).
Let a=eC, so that c=logea.
c2X2 c3x3
ax = eCX = +cx+ + -- +...,
by substituting ex for x in the series (1).
X2 X3.a = 1 + xlog a + (loge a)2 + (loge a)+... ad inf.
2....... 274) t
254. It can be shewn (as in C. Smith's Algebra, Art. 274) that the
series (1), and therefore (2), of the last article is convergent for all real
values of x.