Integration of Power Series

Definition

If a function $f(x)=\sum _{n=0}^{\infty }{a}_n(x-c{)}^n$ converges on $[a,b]$, then $f\in \mathcal{R}[a,b]$, and ${\int }_a^{b}f(x)dx=\sum _{n=0}^{\infty }{a}_n{\int }_a^b(x-c{)}^{n}dx$ where $\mathcal{R}[a,b]$ is a set of integrable function on a $[a,b]$

Improper Integral of Power Series

If a function $f(x)=\sum _{n=0}^{\infty }{a}_n(x-c{)}^n$, and the power series $\sum _{n=0}^{\infty }\frac{a_n}{n+1}(b-c{)}^{n+1}$ converges on $[a,b)$, then $f$ is improper integrable on the $[a,b)$, and ${\int }_a^{b}f(x)dx=\sum _{n=0}^{\infty }{a}_n{\int }_a^b(x-c{)}^{n}dx$