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.

80

Calculation of functions by infinite series.

Bk. I ch. VIII.

1+ i (n - (n-1) X2 +. (. - 1)... (,-n +2) 1...)
1 2- Ix + -'
The error which is incurred in breaking off the series at the ntli term
is at most equal to the maximum value of the remainder
mXn xn (1 + x)m-n or mn Xn,
according as x < or > 0.
It appears from what has been said that, except for positive
integer values of m, for which the series is finite, the values of the
terms of the series for x > + 1 or < - 1 increase beyond any limit,
so that the series no longer converges.
The limiting cases: x - + 1 or = 1 require a special consideration, not at present possible by means of the remainder, inasmuch
as it should take account of the maximum value of (1 - 0)n-1 (1 + O x)?-1-'.
It is plain at once, that if the series converge at all for these limits,
the values it must express are respectively 2m and Om; for, as long
as the series converges, it is a continuous function of x and must
therefore assume the same value as the continuous function (1 + )111
with which it coincides for all values of x within these limits.
When m > 0 and x = - 1, the series is of the form:
* m rn(m-i () (m —1)(m —'2) + ----
1- ( + r( -1) ( - 2) 1)n re(n - 1)... (m —n+1)
1 1' 1 3 + 1I)
In this series the terms all take the same sign as soon as n becomes
> m. But the sum of 2, 3,. n +- 1 terms is
n- 1 (m- 1) (mn-2) (im-l) (m —2) (m -3) (m-1) (-2). (-)
1' - 1 _ —.(), (rn-i) (rn-2)...(r-n )
12. 13 In '
Here each term is ultimately less in absolute amount then the one
preceding; therefore we have a series of numbers all of one sign
and each smaller than the one before it. This series of numbers has
therefore a determinate limit and this limit is zero in consequence of
the above remark.
For x = 1 we obtain the series,m i (m- 1) n (n - 1)(m-n ) r(in m- )(-m —2) *.(1m-n- l)
1-1 _ - _ +...
1+ + T + L +" L '"
These terms assume alternate signs when n > m but yet the series
converges absolutely, because according to what we have just seen, the
series converges, even when we give all its terms the same sign. The
series expresses the value 28n.
When in = - t < 0, the series cannot converge for x =- 1, for
we have (1 -1)-Y = oo. Accordingly if the series converge at all
for x + 1, it can do so only conditionally. We obtain:
*) Newton in the letters for Leibnitz of 13 June and 24 October 1676.