# Statement(s) > [!NOTE] Statement 1 (Integral of Uniformly Convergent Series of Real Functions equals Series of Their Integrals) > Let $(f_{k}),f_{k}:[a,b]\to \mathbb{R}$ be a sequence in [[Riemann integration|Riemann integrable functions]]. Suppose the partial sums $S_{n}:= \sum_{k=1}^{n} f_{k}$ [[Uniform Convergence of Series of Real Functions|converges uniformly]]. Then $\sum_{k=1}^\infty f_{k}$ is Riemann Integrable and $\int \sum_{k=1}^{\infty} f_{k} = \sum_{k=1}^{\infty} \int f_{k}.$ # Proof(s) **Proof of statement 1:** It follows from [[Linearity of Riemann integration|additivity of the Darboux integral]] that each $S_{n}$ is integrable. Since $S_{n}$ converges uniformly, it follows from [[Uniform Convergence preserves Riemann Integrability]] that $\lim_{ n \to \infty } \int S_{n} = \int \lim_{ n \to \infty } S_{n} .$Since $\int S_{n} = \sum_{k=1}^{n}\int f_{k}$ and $\lim_{ n \to \infty } S_{n}=\sum_{k=1} f_{k}.$ $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography