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.

290 Examples on the calculation of definite integrals. Bk. III. ch. VII.

Multiplying out these products, the absolute amount resulting
from the first equation is easily seen to be not less than the absolute amount resulting from the second; therefore we have the relation:
abs -Q+- - 1 > abs [P — 1]
By hypothesis we can choose n so as to make the amount on the left
side arbitrarily small; therefore also a value n can be found, such that
abs [P,+k i] shall be < 6.
Hence follows that P, tends to a determinate finite limiting value that
is not zero. For were zero the limiting value of P, to each value
of n however great could be found a value k, for which the ratio
Pn+k: P, would be arbitrarily small.
A necessary and sufficient criterion for the absolute convergence
of the product:
P=(l -- u1) (1 l+ t2) (1 1+ t)..,
is the convergence of the infinite series formed of the absolute amounts:
vr + V-, + t- 3 + * * -+ V +.. ~ etc..
For, because the product:
Qn = (1 + V) (1 +.) * ' * + Vn) is > V1 + 2 + * *...n
the convergence of the infinite series is necessary, and because:
V1 c- I + < 1 v < e, 1 + 1 n e
so that:
Qn is < evi+2+.+..-n
therefore the convergence of the infinite series is sufficient.
The value of an absolutely convergent product is independent of
the arrangement of its factors.
For, writing:
PI = (1 + qtl) (1 + U)... (12 + n,),
P,,'= (1 + <l') (] + ug')... ( + r,,,),
this second product consisting of factors occurring in P only arranged
differently, we can choose m so large, that all the factors contained in
P, shall also occur in Pm'. Then:
Pn -' Pn (1 + Uk) (1 + I)... (1 + tU),
where k,... v denote indices that are greater than n; or:
-n (1 k) (1 + u6).o (1+ t v)
n
But the amount of the right side is not greater than the amount of
(1 - Vk) (1+ v-l)... (I + v- ) and this is less than
evk +l- +"+.. %.