The questions asks...
$$\textrm{For what values of }x \textrm{ do the following series converge, where }a > 0,\, b > 0?$$
I have managed to find the correct values of $x$ for the first two parts, but the final series is stumping me. The series is...
$$\sum\frac{x^n}{a^n+b^n} \textrm{, which becomes } \frac{a^n+b^n}{a^{n+1}+b^{n+1}}x\,\,\textrm{ after doing the ratio test}$$
I understand that if the fraction tends to $0$ then it works for all values of $x$, but I'm not sure how to prove anything beyond this point.
Hint: Prove that$$\lim_{n\to\infty}\frac{a^n+b^n}{a^{n+1}+b^{n+1}}=\frac1{\max\{a,b\}}.$$