 # Radius of convergence of power series

We look here at the radius of convergence of the sum and product of power series.

Let’s recall that for a power series $$\displaystyle \sum_{n=0}^\infty a_n x^n$$ where $$0$$ is not the only convergence point, the radius of convergence is the unique real $$0 < R \le \infty$$ such that the series converges whenever $$\vert x \vert < R$$ and diverges whenever $$\vert x \vert > R$$.

Given two power series with radii of convergence $$R_1$$ and $$R_2$$, i.e.
\begin{align*}
\displaystyle f_1(x) = \sum_{n=0}^\infty a_n x^n, \ \vert x \vert < R_1 \\ \displaystyle f_2(x) = \sum_{n=0}^\infty b_n x^n, \ \vert x \vert < R_2 \end{align*} The sum of the power series \begin{align*} \displaystyle f_1(x) + f_2(x) &= \sum_{n=0}^\infty a_n x^n + \sum_{n=0}^\infty b_n x^n \\ &=\sum_{n=0}^\infty (a_n + b_n) x^n \end{align*} and its Cauchy product:
\begin{align*}
\displaystyle f_1(x) \cdot f_2(x) &= \left(\sum_{n=0}^\infty a_n x^n\right) \cdot \left(\sum_{n=0}^\infty b_n x^n \right) \\
&=\sum_{n=0}^\infty \left( \sum_{l=0}^n a_l b_{n-l}\right) x^n
\end{align*}
both have radii of convergence greater than or equal to $$\min \{R_1,R_2\}$$.

The radii can indeed be greater than $$\min \{R_1,R_2\}$$. Let’s give examples.

### Sum of power series

Consider any power series $$\displaystyle f_1(x) = \sum_{n=0}^\infty a_n x^n$$ having a non-zero finite radius of convergence $$R_1$$. Then the radius of convergence of the power series $$\displaystyle f_2(x) = -f_1(x) = \sum_{n=0}^\infty -a_n x^n$$ is also equal to $$R_1$$. The sum $$f_1(x) + f_2(x)$$ is the always vanishing power series whose radius of convergence is infinite, hence greater than $$R_1$$.

### Product of power series

Here we take
\begin{align*}
\displaystyle f_1(x) &=\frac{1+x}{1-x} = (1+x) \left(\sum_{n=0}^\infty x^n\right)\\
&= \sum_{n=0}^\infty x^n + x \sum_{n=0}^\infty x^n\\
&=1 + 2 \sum_{n=1}^\infty x^n, \qquad \vert x \vert < 1 \end{align*} with radius of convergence $$R_1=1$$. And $$f_2(x)=f_1(-x)=\frac{1-x}{1+x}$$. We have $\displaystyle f_2(x)= 1 + 2 \sum_{n=1}^\infty (-x)^n$ The radius of convergence of $$f_2(x)$$ is also equal to $$1$$. The product power series is $f_1(x) f_2(x) = \frac{1+x}{1-x} \cdot \frac{1-x}{1+x}=1$ and has an infinite radius of convergence. For that matter, one can directly verify that the Cauchy product of $$f_1(x)$$ and $$f_2(x)$$ has all coefficients vanishing except the first one which is equal to $$1$$.