Plane trigonometry, by S.L. Loney.

THE QUANTITY e.

297

Hence it lies between 2 and 3.
By taking a sufficient number of terms in the series, it
can be shewn that
e = 2'7182818285...
252. The quantity e is incommensurable.
For, if possible, suppose it to be equal to a fraction -, where p and q
are whole numbers.
We have then
-=1I+ -....+.........(1).
Multiply this equation by q, so that all the terms of the series (1)
become integers except those commencing with q. Hence we have
p Il- 1 =whole number+ -+ + +...
1 1 1
i.e. an integer= + + + (2).
i.e., an integer=q-~+ + (q + 1) (q + 2) + (q + 1) (q + 2) (q + 3)
But the right-hand side of this equation is > I, and
1 1 1
c +3 +
q+l (q+1)2 (q+1) '"
i.e. is < +1 (1 - 1,
~~.. 1
i.e. is <-.
q
1 1
Hence the right-hand side of (2) lies between and -, and is thereq +1 q'
fore a fraction and so cannot be equal to the left-hand side.
Hence our supposition that e was commensurable is incorrect and it
therefore must be incommensurable.
253. Exponential Series. When x is real, to prove
that
X2 X3
e- =1 + x ~ +... ad inf.
e X F1+ T +