A course of pure mathematics, by G.H. Hardy.

195] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 363
195. The logarithmic series. Another very important
expansion in powers of x is that of log(l +). Since
log(1 +)= i t'
and 1/(1 + t)= 1 - t + t2 -... if t is numerically less than unity, it
is natural to expect* that log(l + x), when - 1 <x < 1, will be
equal to the series obtained by integrating each term of the
series between 0 and x, i.e. to the series x - X2 + 3 -.... And
this is in fact the case. For
1/(1 + t= 1 - t + t2-... + (-1)m-1 + {(- l)'nt/(l + t)}
and so, if > - 1,
log (1 X+ x):f:dt+ (- 1 ) m\og(~+x)^ _x-: +.+. +(-1)yR'R,
jx in Mn
where J ftmt
-Rm = 1 +t'
We require to show that the limit of Rm, when m tends to oo, is
zero. This is almost obvious when x is positive and less than or
equal to unity; for then Rm is positive and less than
Jorx (xm+1)
f tmdt = __1_
and therefore less than 1/(m + 1).
If -1 x< 0 we put t=-u and x=-a, so that
t / [ du
Rm = ( -1) 1
1 -- 'itJ
which shows that Rm has the sign of (- l). Also, since the
greatest value of 1/(1 - u) in the range of integration is 1/(1 -),
we have
1 ______ __ 1
I Rm 1-^o (m+1)(1-:) <(m +l)(l- )
and so lim Rn = 0. Hence
log (1 + 7) = x - x2 + 3 -...,
provided- 1 <x 1. If x lies outside these limits the series is
not convergent. If x= 1 we obtain
log2=-1 +-...,
a result already proved otherwise (Ex. xcI. 6).

* See Appendix II for some further remarks on this subject.