Help with radius of convergence of a power series. I need to determine the radius of convergence of the series $\sum_{n=1}^\infty a_nx^n$, where $a_n=a^n+b^n$ and $a,b$ are real numbers. Not sure how to approach this one.
 A: By Cauchy's-Hadamard formula, with $\;R:=$ convergence radius, with the usual conventions when $\;R=0\,,\,\infty\;$ , we get:
$$\frac1R=\lim_{n\to\infty}\sup\sqrt[n]{|a^n+b^n|}$$
and assuming $\;|a|\ge|b|\;$ , we get
$$\sqrt[n]{|a^n+b^n|}=|a|\sqrt[n]{1+\left(\frac{|b|}{|a|}\right)^n}\xrightarrow[n\to\infty]{}|a|$$
A: Since $a_n = a^n + b^n$, then
$$\sum_{n=1}^\infty a_n x^n =\sum_{n=1}^\infty (ax)^n + \sum_{n=1}^\infty (bx)^n .$$
For this to be convergent, both series must be convergent, but these are regular geometric progressions, so the conditions for their convergence are that $|ax|<1$, and $|bx|<1$. So we must have simultaneously 
$$|x|< \frac{1}{|a|}$$
and
$$|x|<\frac{1}{|b|}$$
Which is equivalent to $$|x| < \frac{1}{\max(|a|,|b|)}.$$
A: Your series converges when
$$\lim_{n \to \infty}\left\vert\sqrt[n]{\left \vert a^n+b^n\right \vert} x \right\vert < 1$$
Hence, the radius of convergence is
$$R = \dfrac1{\lim_{n \to \infty} \sqrt[n]{\left \vert a^n+b^n\right \vert}} = \dfrac1{\max(\vert a \vert, \vert b \vert)}$$
