Let all of the functions

(1)

with ,
1, 2, ..., be regular at least for , and let
be uniformly convergent for for every . Then the coefficients in any column form a convergent
series. Furthermore, setting

(4)

for ,
1, 2, ..., it then follows that

(5)

is the power series for , which converges at least for .
See also
Double Series
Explore with WolframAlpha
References
Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York:
Dover, p. 83, 1996.Referenced on WolframAlpha
Weierstrass's Double
Series Theorem
Cite this as:
Weisstein, Eric W. "Weierstrass's Double Series Theorem." From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrasssDoubleSeriesTheorem.html
Subject classifications