Convergence of Real Power Series

Convergence

Definition: Convergence and Divergence of Power Series

Let $\sum _{n\in \mathcal{D}}{a}_n(x-c{)}^n$ be real power series and let $x^{\ast }\in \mathbb{R}$.

We say that $\sum _{n\in \mathcal{D}}{a}_n(x-c{)}^n$

• is convergent or converges for $x^{\ast }$ if the resultant real series ${\sum }_{n\in \mathcal{D}}{a}_n(x^{\ast }-c{)}^n$ is convergent;
• is absolutely convergent or converges absolutely for $x^{\ast }$ if the resultant real series ${\sum }_{n\in \mathcal{D}}{a}_n(x^{\ast }-c{)}^n$ is absolutely convergent;
• is divergent or diverges for $x^{\ast }$ if the resultant real series ${\sum }_{n\in \mathcal{D}}{a}_n(x^{\ast }-c{)}^n$ is divergent.

Interval of Convergence

Theorem: Interval of Convergence

There are only three possibilities for the convergence of every real power series $\sum _{n\in \mathcal{D}}{a}_n(x-c{)}^n$:

• The power series converges only for $x=c$.

• The power series converges for all $x\in \mathbb{R}$.

• There exists some $r>0$ such that the power series converges if $x\in (c-r;c+r)$ and diverges if $|x-c|>r$. In this case, the power series may or may not converge for $x=c+r$ or $x=c-r$ or both

PROOF

TODO

Definition: Interval of Convergence

The set of all $x\in \mathbb{R}$ for which a power series $\sum _{n\in \mathcal{D}}{a}_n(x-c{)}^n$ converges is known as its interval of convergence and if this interval is finite, then we call half of its length the radius of convergence.

Determining the Interval of Convergence

Algorithm: Determining the Interval of Convergence

We are given a real power series $\sum _{n\in \mathcal{D}}{a}_n(x-c{)}^n$ and want to determine its interval of convergence.

1. Evaluate either one of the limits ${\lim }_{n\to \infty }\left|\frac{a_n}{a_{n+1}}\right|$ and ${\lim }_{n\to \infty }\frac{1}{\sqrt[n]{|a_n|}}$. Choose whichever one is easier to calculate.

2. If the limit is zero, then the power series converges only for $x=c$.

3. If the limit is $+\infty$, then the power series converges for every $x\in \mathbb{R}$.

4. If the limit is equal to some nonzero $r\in \mathbb{R}$, then the power series converges for $x\in (c-r;c+r)$. However, we also need to check whether it converges for $x=c-r$ and $x=c+r$.

• Evaluate the power series at $x=c-r$. If the resultant series ${\sum }_{n\in \mathcal{D}}{a}_n(-r{)}^n$ is convergent, then the power series converges for $x=c-r$ as well.
• Evaluate the power series at $x=c+r$. If the resultant series ${\sum }_{n\in \mathcal{D}}{a}_n{r}^n$ is convergent, then the power series converges for $x=c+r$ as well.