Plane trigonometry, by S.L. Loney.

EXPONENTIAL THEOREM. 299
255.. 1. Prove that (e- = + + +... ad inf.
By equation (1) of Art. 253 we have, by putting x in succession equal
to 1 and -1,
1 1 1 1
e=l ++ + +3 +-... ad inf.
and e-li = 1 +- + -... ad inf.
Hence, by subtraction,
e-e-l=2 1+5+ -)'
1 1 1\,1 1
i.e. I e- )=l+ 15+
e - =+ +...
Ex. 2. Find the sum of the series
1+2 1+2+3 1+2+3+4
1+ -2 + - +... ad inf.
12 13 14
1+2+3+...+n 2n(n1l)
The nth term =
In In
1 n+1 l(n-1)+2] 1 2.2 n- 2 L |I-l 2 ln-2 n-1i'
provided that n > 2.
Similarly
the (n - )th term= L + 2]
the 4th term= [1 + 2
the 3rd term= [ + ].
Also the 2nd term= 1 + 2,
and the 1st term = [ 2.