# Dini theorem

From Encyclopedia of Mathematics

*on uniform convergence*

If the functions $u_n$, $n=1,2,\ldots,$ are continuous and non-negative on a segment $[a,b]$ and if the sum of the series $\sum_{n=1}^\infty u_n$ is a continuous function on this segment, then the series converges uniformly on $[a,b]$. Dini's theorem can be generalized to the case when an arbitrary compactum is the domain of definition of the functions $u_n$.

**How to Cite This Entry:**

Dini theorem.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Dini_theorem&oldid=32779

This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article