Radius of convergence of $2^n+3^n, n \geq 1$

Question

Find the radius of convergence of the power series where, $a_n= 2^n+3^n, n \geq 1$. The answer is given to be 1.

The tests I can use are Cauchy Hadamard Test and Ratio Test.

My attempt: Using Ratio Test:

$a_{n+1}= 2^{n+1}+3^{n+1}$,

which is giving me $|\frac{a_{n+1}}{a_n}|= \frac {\frac{1}{3} (\frac{2}{3}+(\frac{3}{2})^n)}{1+(\frac{3}{2})^n}$

So I'm not going anywhere in this method.

Using Cauchy Hadamard, I failed to find the limsup. Please help!

Answer 1

By the ratio test we get

$$\frac{a_{n+1}}{a_n}=3\frac{\left(\frac23\right)^{n+1}+1}{\left(\frac23\right)^{n}+1}\xrightarrow{n\to\infty}3\implies R=\frac13$$

Remark Notice that if $\sum a_n x^n$ and $\sum b_n x^n$ are two power series with radius of convergence $R_a$ and $R_b$ and if $R_a\ne R_b$ then the radius of convergence of $\sum (a_n+b_n)x^n$ is $\min(R_a,R_b)$.