Plane trigonometry, by S.L. Loney.

300

TRIGONOMETRY.

Hence, by addition, the whole series
2 1+ + l + +1 adinf.]
+.2[1+l + +_+... adinf.]
1 3e
=2'e+e= 2
2 2
256. Logarithmic Series. To prove that, when y is
real and numerically < 1, then
1 1 1
lo ( + y) = y- 2 + Y3 - 4 +...ad inf.
In the equation (2) of Art. 253, put
a = +y,
and we have
x 2
(1 + y)x =1 + x log, (1 + y) + 1{2 loge (1 + y)2 +...(1).
But, since y is real and numerically < unity, we have
2(1 + - 1) X+ (x- 1) ((x- 2)s+
(l+^y)=l^.^^.y~^ 31.2 1.2.3............... (2).
The series on the right-hand side of (1) and (2) are
equal to one another and both convergent, when y is
numerically < 1. Hence we may equate like powers of x.
Thus we have
( =_ (- ) (- 2) f (-I) (-2) (-3)
(I = -1.2+ 1. 2.3. 2. 3. 4 Y4
+... ad inf.,
1 +y1 1y2+ y3_ 1
i.e. log (1 + y) = y - yt + y3 y4 +......(3).