-16-

o
absolutely convergent) and therefore 5_—, is a power series con-

verging absolutely for |x| < Ry, and uniformly for |x| £R; <R, .

2 8
Now ¥ = 1+u+.;_1.+72’1.'.+-~--+§+"" is a series aof

power series satisfying the Weierstrass theorem, so that it may be
arranged according to powers of x. After this rearrangement is made
the resulting series is convergent in Ry. The expression for the
ccefficient of x® will be denoted by C,, and is found by tsking the
sum of all the terms, chosen from e¥, such that Sngh; = w , These

terms having been obtained by the use of (1C), the sum of the coef-

ficients of a%, that is, C,, is
an Imny oMy Mg g
s tel) ml " mgl "ttt mgl Manks oy g
C- =7 (sj1h1°°'rl) (Sj'k’-o.rq) )

ol Ml vee gl Jif Tocae gr'?

where the svmzation is tc be taken for all the values of the

indices for which %'fl‘h¢ = w ; and where bty definiticn,

hi=ja+ kf 4+ +rd,and mi=jg+ kit 4r;=1, and Ji = jlleop!
Using this expression for C, , we have

IPLE ' wgle)) w1 4 Can 4 Cax®  2ronseiyf ng¥y wesenss

The series for e¥ converges for all values of u, but since we know
cnly that u converges for |x| < R,, then R, is -less than or equal
to the radius of convergence cf the series representation of i, We
have proved the following theorem:

TueorsM I11la. Let the eclements of the array (1) be given
such thet II[1 + v¢(x)] converges uniformly nhen O < |x| <R,, where
vy(x) is defined on page 7, and uhere |vy(x)| <1~-c¢, (C<e<1); and
let (1) also be given so that all the s's converge. Then for
|x] € Re < Ry the groduct II and the series 1 + chxw are
equivalent, where the C's are given by formula (11).

The theorem can be stated with a different hypothesis:

TueorEw IIIb. Zet the elesents of the array (1) be given
such that |vy(x)| <1 -6, (C<e<1), where vi(x) satisfies the definition