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.

68

Calculation of functions by infinite series.

Bk. I ch. VIII.

less than an arbitrarily small quantity. The calculation of the number
e affords us an example of this. The formula of such an infinite sum or
of such an infinite product is called a convergent one.*)
39. The property of convergence of any infinite series is expressed
analytically as follows:
Let Ul + Ut- + t3 + ' * * t + utn+ + Ut+2 + * *
be the terms of the infinite series, which can be continued unlimitedly
according to some law, the sums obtained by adding up, first n terms,
then n + 1, n + 2, -. n -+ terms:
Sn, = t 1 + '2 + * * ' + n
Sn+l =- u1 + t2 + * * * + utt + ZU1+1.......................
Sn+k = -1 + '.2 + * * * + qn + Un+1 - + -f- u+ Z
must form a succession of numbers with a determinate finite limiting
value S. This requires: first, that none of these sums, therefore
also none of the terms u, increase beyond any finite amount, and
second, that for any number Y however small, a place n be assignable,
such that the amount of the difference Sn+k —Sn for every value of ],
shall be less than 6. But this difference is nothing else than the
sum of k terms following the nth; accordingly it must be possible to
choose n so that for every value of k,
abs [Lun- + -n+2 + u- +J q k] shall be < 6 -Now let us denote by R, the difference between the finite limiting
value S and the sum S,, then this quantity S,,+k - S may also be
written as Rn -- -Rn+k, similarly Sn+k- Sn+l as Rn+i - Rn+k, etc.,
whence, provided the choice of n makes abs [Rn, - ln+k] less than 6
for every value of D), Rn+i - Rn+k R-n+2 -- Rn+k, etc. are also certain
to remain less in amount than 26 for every value of k. For we have
abs [Rn+, -Rn+k] = abs [(Sn+k - - Sn)] (Sn+ Sn)]
abs [Rn+2 -- n+k] = abs [(Sn+k - Sn) -- (+2 -- Sn)], etc.
And conversely, when Rn, Rn+i,, are smaller than 6, the differences
Rn - -Rn+k =: /- n+1 + Un+2 *.* - + zUn-k = Sn+k - Sn
are also less than 2 4 for every value of k.
If then we call the limit of the sum R,, of all terms from the (n + 1 )th
*) It is important for the beginner to realise clearly this requisite of calculability of a function; and so the essential difference between a rational
function and all other denominations of functions -, log, sin, cos. The latter
are only to be regarded as symbols by which the dependence of one number on
another is expressed, whose properties are no doubt known, so that the nature of
the dependence is completely defined ex. gr. by inversion of an arithmetical
operation, or by geometric definitions, but for whose calculation we have as yet
no fixed law.