Convergence of a series of functions. Notation/ idea

Question

When we speak of a series of functions, $$\sum_{k = 1}^{\infty}f_{k}(x)$$, converging. We are saying that the series is converging to a specific function correct? I.e: $$\sum_{k = 1}^{\infty}f_{k}(x) \rightarrow f(x)$$ or is it to a series of functions ? i.e: $$\sum_{k = 1}^{\infty}f_{k}(x) \rightarrow \sum_{k= 1}^{\infty}f(x)$$ (This one doesn't seem correct because all you would get in the end is $$nf(x)$$.

Answer 1

When we write $\sum_{k=1}^{\infty}f_k$, we actually mean the sequence $\{\sum_{k=1}^nf_k\:|n \in \mathbb{N}\}$, and if the $\lim_{n\rightarrow \infty}\sum_{k=1}^n f_k(x)$ exists for all $x$ in the domain, then we define $\sum_{k=1}^{\infty}f_k(x) := \lim_{n\rightarrow \infty}\sum_{k=1}^n f_k(x)$. It is important to note that the sequence and the limit has the same notation (which can be rather confusing), but the limit is indeed a function.

Saying that $"\sum_{k=1}^{\infty}f_k(x)\rightarrow f(x)"$ is the same as saying that $"\lim_{n\rightarrow \infty}\sum_{k=1}^{n}f_k(x) \rightarrow f(x)"$, which is a weird way of saying that $\sum_{k=1}^\infty f_k(x)=\lim_{n\rightarrow \infty}\sum_{k=1}^{n}f_k(x)=f(x)$. Also if the limit does exist, then by definition we have $\sum_{k=1}^{n}f_k(x) \rightarrow \sum_{k=1}^{\infty}f_k(x)$.

Answer 2

The series would converge to a single function $f(x)$. Otherwise, you can't really say it converges.