Plane trigonometry, by S.L. Loney.

364 TRIGONOMETRY.
X2 X3
Then e = 1 +-x+- + +......ad inf.
r2 (cos 20 + i sin 20)
=1 ~ r(cos 0+ i sin 0) +r2 (cos 20 + i sin 2)
= - r (cos 3 + i sin 0) +i 3
+-' +......adinf.
r2 cos 20 r3 cos 30
= +rcos 0 + 2 + — +......
I-r2 sin 20 r sin 30
+J-1 rsin+ 12 + -- - +......
L r + +
The quantity
r2 r3
1 + rcos0 + 2 cos 2 + L cos 30 +......
r2 r3
is <1+r+ + +......
and is therefore convergent since this series is convergent
for all real values of r. (Art. 254.)
Similarly the quantity
r sin - + ~ sin 20 +......
is convergent.
Hence the series for el is always convergent.
304. When x is a complex quantity the quantity ex
is then a short way of writing
x2 X 3
1 +X+ 2+ +.3......
Unless x be real, the e in ex does not mean the series
1+ 1+ 1
1+1+]2++ $-.......