An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

82

Calculation of functions by infinite series. Bk. I ch. VIII.

The first form of the remainder shows that the series still converges
for x = 1.
Therefore
X2 X3 24 Xn
(1 + x) x- 2 + 3 4+. (- )-' f... l < <+
In the particular case
(2) = 1 - + — - + -.
of this last, we have an example of a merely conditionally convergent
series; for neither the series
1 + 2 +~ 1 + I-T + 1 ~.,
nor the series
l + I + o1 +.. * *4, nor 2 + 8 + - + 8- 7
are convergent, although their terms decrease and converge to zero,
on the contrary, their sums increase beyond any finite amount; for
+ I 3 1 + I1 > 1 l + i I + I >,..
1 1+
2 +1 + +2 + 1 n-t > 1, etc. ).
2n+1 2?-c+-2 2f+~ 2
To obtain series useful for calculating the logarithm of any positive
number, let us put - x for x in the series just found, then if x < 1
x2 X3 _ _ __ Pt
2 3 4: n —I
l(1 - x) -- x- 2 - 3 a —4 '-1 + ',
therefore
71 + X = ( 1, s X9, 2k+1
~ + x_2 +-+...,~k+ /+ --
X + 3 + 5- 2k+l)
and as R - R' converges to zero in the assumed interval, we have
*) The above divergent series 1 + ~ + - j- +. is called the harmonic
series. It is important for subsequent applications to remark that the series
1 1 1 1 1
1 1
+t __ 1+r +
converges for all values of, > 1. For, grouping as above =..
2M 3) 2L 2 I-'
4+..+ t2L- + < 4. = ( —l_,
4At 5V 6Y 7"11 44 \2
1 1 I / 13
1+... -- <t-h- ( t1..... *.............
we see, that the sum of any number of terms of the series remains less than the
sum of the same number of corresponding terms in the geometric progression
whose ratio is the proper fraction2~-1