654 
f@=f,@+f,@) +... 
be quasi-uniformly convergent in the interval a<a<)b. 
f(@) is then continuous in the latter open interval. Let M and m 
denote the maximum and minimum of f(x) at a and let u be an 
arbitrary number of the interval m<u< M. A set of points 2,, a,, 
&,..-2,,... can be constructed where lm 2, =a and lim f(z‚)=u. 
yo yo 
From a limited number of indices it is possible to choose one for 
each point of this set such that |S, (7,)—/f(a)| <<, where e, is an 
arbitrary positive number. There are therefore an infinite number 
of points a where one and the same index can be used, which we 
call n,. If 2, tends to a, then S,,(a,) tends to S,,(a) and f(x) to u; 
benee |S,,(a)—wu| Se. Let ¢,,¢,,... be a decreasing sequence of 
positive numbers having zero for limit. It is again possible to choose 
from a finite number of indices for every z, an index n, >> n, such 
that |S, («,)—f(#,)| <€,, hence there exists an index n, satisfying 
Sd) ul Se, 
Thus pursuing we find that there is a partial sequence of functions 
Sn(@), Sn(@),... which at a converges to the value u and for a <a Sb 
to f(z). Hence: 
If the series f,@) + f‚(@) +... consists of terms which are 
continuous within a<xa<b and converges quasi-uniformly to f(a 
in a<x<b, and if w is an arbitrary value lying between the 
maximum and the minimum of f(x) at a, then the series can be 
transformed, by uniting the terms qroup-wise to one new term, into 
another series which converges na<a<b, having u for its limit 
at a and f(x) at the other points. 
In the example of § 5 we have 1/—m=0O. The series (aa) + 
+ (w?—w’) +... is here a transformed series which converges every- 
where to zero. 
In the example of § 6 M—1, m=O. Choose a set P,,, Pi, 
having u as limit.. The partial sequence S,,(z), S,,(#),... converges 
to f(x) for O< «<1 and tou at 0. 
That the quasi-uniformity of the convergence in the open interval 
is no superfluous condition is illustrated by the series 1—a-+ x?—a*-+.., 
which for 0< «<1 represents = so that for «=1 we have 
M=m= }. In no way however the terms of the series 1—1-+1—1-... 
can be united to groups in order that the transformed series should 
converge to 4. 
8. The theorem of the preceding § can be reversed as follows: 
