# Definition(s) > [!NOTE] Definition 1 (Series of Real Functions) > Let $(f_{n})$ be a [[Sequences|sequence]] of real functions $f_{k}:\Omega \subset \mathbb{R}\to \mathbb{R}.$ Let $(S_{n})$ be the sequence of partial sums, with $S_{n}:\Omega\to \mathbb{R}$ defined by $S_{n}(x)=\sum_{k=1}^{n} f_{k}(x)$Then we say that the series $\sum_{k=1}^{\infty} f_{k}(x)$converges pointwise to $S:\Omega\to \mathbb{R}$ in $\Omega$ if $S_{n}$ [[Pointwise Convergence of Sequence of Real Functions doesn't Imply Uniform Convergence|converges pointwise]] to $S.$ > [!Example] Example > Contents # Properties(s) # Application(s) **More examples**: # Bibliography