Convergence of Power Series

Theorem

Consider the power series ${\sum }_{n=0}^{\infty }{c}_n(x-a{)}^n$. The series satisfies exactly one of the following properties:

i. The series converges at $x=a$ and diverges for all $x\ne a$.
ii. The series converges for all real numbers $x$.
iii. There exists a real number $R>0$ such that the series converges if $|x-a|<R$ and diverges if $|x-a|>R$. At the values $x$ where $|x-a|=R$, the series may converge or diverge.