[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

Prove that every power series can be integrated, term by term. Specifically, prove the following two theorems.

• A power series [math]\sum_{i=0''^\infty a_i(x-a)^i[/math] and its integrated series
  [[math]] \sum_{i=0}^\infty \frac{a_i}{i+1} (x-a)^{i+1} [[/math]]
  have the same radius of convergence.}
• If the radius of convergence [math]\rho[/math] of the power series [math]\sum_{i=0''^\infty a_i(x-a)^i[/math] is not zero and if [math]f[/math] and [math]F[/math] are the functions defined, respectively, by
  [[math]] f(x) = \sum_{i=0}^\infty a_i(x-a)^i \quad \mbox{and} \quad F(x) = \sum_{i=0}^\infty \frac{a_i}{i+1} (x-a)^{i+1} , [[/math]]
  then
  [[math]] F(x) = \int f(x) \; dx + c . [[/math]]
  }