# Question on a proposition of uniorm convergence of function series

The Proposition states if $$(f_n)_{n\in\mathbb{N}}$$ is a secuence of functions $$f_n:D\rightarrow \mathbb{C}$$ with $$\sum_{n=1}^{\infty} ||f_n||_D < +\infty$$ then the series $$\sum_{n=1}^{\infty}f_n$$ converges absolutely uniform and pointwise.

The series converges uniformly (or uniformous?)if and only if the secuence of partial sums do so. The series of partialsums is $$(\sum_{i=1}^{n}f_n)_{n\in\mathbb{N}}$$

A sequence of functions converge if and only if

$$\exists_{f:D\rightarrow\mathbb{C}}\forall_{\epsilon>0}\exists_{n(\epsilon)\in\mathbb{N}}\forall_{n\geq n(\epsilon)}\forall_{z\in D}:|f_n(z)-f(z)|<\epsilon$$

Which would mean

$$\exists_{f:D\rightarrow\mathbb{C}}\forall_{\epsilon>0}\exists_{n(\epsilon)\in\mathbb{N}}\forall_{n\geq n(\epsilon)}\forall_{z\in D}:|\sum_{i=1}^{n}f_n(z)-f(z)|<\epsilon$$

My Question what would be the $$f$$ in this Proposition?

• Perhaps you are having a translation difficulty (English is an immensely irregular language). "Uniformous" is not a word in English, although logically it could be. The correct word is "uniformly". – DanielWainfleet Jan 10 at 9:28

For each $$z$$ the series $$\sum\limits_{k=1}^{\infty} f_k(z)$$ is absolutely convergent. $$f(z)$$ is the sum of this series. Note that $$| \sum\limits_{k=1}^{n} f_k(z) - \sum\limits_{k=1}^{m} f_k(z) |<\epsilon$$ for all $$z \in D$$ for all $$n,m \geq n_0$$ implies $$| \sum\limits_{k=1}^{n} f_k(z) -f(z) |\leq \epsilon$$ for all $$z \in D$$ for all $$n \geq n_0$$.