Integration of Sequence of Functions

Definition

Integration of Sequence of Functions

If $\forall n\in \mathbb{N},f_n\in \mathcal{R}[a,b]$, and $(f_n)$ converges uniformly to $f$ on an interval $[a,b]$, then $f\in \mathcal{R}[a,b]$, and $\underset{n\to \infty }{\lim }{\int }_a^b{f}_n(x)dx={\int }_a^{b}f(x)dx$ where $\mathcal{R}[a,b]$ is a set of integrable function on a $[a,b]$

Term by Term Integration

If $\forall n\in \mathbb{N},f_n\in \mathbb{R}[a,b]$, and $\sum _{n=1}^{\infty }{f}_n$ converges uniformly to $f$ on $[a,b]$, then $f\in \mathcal{R}[a,b]$, and ${\int }_a^{b}f(x)dx=\sum _{n=1}^{\infty }{\int }_a^b{f}_n(x)dx$ where $\mathcal{R}[a,b]$ is a set of integrable function on a $[a,b]$