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''for uniform convergence of series of functions''

If a sequence of real-valued functions  $  a _ {n} ( x) $, 
$  n = 1, 2 \dots $
is monotone for every  $  x \in E $, 
where  $  E $
is a certain set, and converges uniformly to zero on  $  E $, 
and if the sequence of partial sums of a series  $  \sum _ {n = 1 }   ^  \infty  b _ {n} ( x) $
is bounded on  $  E $(
the functions  $  b _ {n} ( x) $
may take complex values), then the series  $  \sum _ {n = 1 }   ^  \infty  a _ {n} ( x) b _ {n} ( x) $
converges uniformly on  $  E $.

This criterion was established by G.H. Hardy [[#References|[1]]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   "Some theorems connected with Abel's theorem on the continuity of power series"  ''Proc. London. Math. Soc. (2)'' , '''4'''  (1907)  pp. 247–265 {{ZBL|37.0429.01}}</TD></TR>
</table>

====Comments====
This criterion can be proved using the [[Euler–MacLaurin formula]] (cf. the proof of Thm. 3.42 in [[#References|[a1]]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin, "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 107–108</TD></TR>
</table>