Plane trigonometry, by S.L. Loney.

296

TRIGONOMETRY.

Hence the limiting value, when n is infinite, of (1 +
is the sum of the series
1 + 1 + + | + +... ad inf.
The sum of this series is always denoted by the
quantity e.
Hence we have
Lt 1 + =e,
n=co n
where Lt stands for " the limit when n = oo."
n=00
Cor. By putting n=, it follows (since m is zero
when n is infinity) that
1 1 n
Lt (1 + m)m =Lt (I + =e.
m=0O n=oo n
251. This quantity e is finite.
1 1 1
For since 1 <
2.2 22'
1 1 1
4 < 2.2.2 <23 '
we have
e < + + +l+3 +......adinf.
1
< +1 -<1+2, i.e. <3.
Also clearly e > 2.