72 Transactions of the Academy of Science of St. Louis 
T.24. If an infinite series converges absolutely, it converges. 
T.25. A derangement of the terms of an absolutely conver- 
gent series does not alter the sum of the series. 
If each of the terms of an infinite series is a function of x, 
we may denote the series by >. “f;(x), where x has some given 
range. We define S,(x) by the equation 
5.) = D fila) 
and if { Sn (x) } converges Sits: then the Sao series is 
said to converge uniformly over the given range. The Weier- 
strass M-test, as expressed in the following theorem is useful. 
T.26. If for all values of x in a given range R, the norms of 
the terms of a series poe fi(x) are respectively less than the corre- 
sponding terms in a convergent series of numbers pe ui, Where 
mi is independent of x, then rot fi(x) is uniformly convergent in 
K. 
Power series. An infinite series of the form 
Qtr) +H Opp + + HF Orpen X™ °° 
is called a power series. For those values of x where the series 
converges there is defined a function on Ep) to Ej,;. We shall 
be interested in the properties of this function, but it is of pri- 
mary importance to determine the range of x for which the se- 
ries has a meaning. This we proceed to do. 
Let # be a unit element of E,,), that is, an element whose 
normis1. The aggregate of values for which x= £&%), & being a 
variable complex number, is called the complex-plane deter- 
mined by #o, or, more simply, the #) complex-plane. On fixing 
the argument of a power series to the complex-plane determined 
by some fixed unit element i, we have the theorem 
T.27. The series ye | @,r+njXo" } " converges uniformly and 
absolutely for all values of € such that 
| €| < Jim later ||, 
and diverges when 
[€| > tim llaram el". 
The right-hand member of these inequalities, which we de- 
note by p-,, is called the radius of convergence in the Xo complex- 
plane. On considering the totality of unit elements in £11), and 
on applying the Weierstrass M-test, we can easily see the truth 
of the following theorem. 
