then find the the radius of convergence of the following power series $\sum_{n=1} ^{\infty}a_nx^n$ about $ x =0$?

Question

let $a_n = \frac {(1+(-1)^n)}{2^n} + \frac {(1 +(-1)^{n-1})}{3^n}$. then find the the radius of convergence of the following power series $\sum_{n=1} ^{\infty}a_nx^n$ about $ x =0$?

My attempts :by Cauchy–Hadamard theorem the radius of convergence will be 1

Answer 1

You can write that $$ a_{2n}=\frac{1}{2^{n-1}} \text{and } a_{2n+1}=\frac{2}{3^n}$$ You can see that $$ \frac{a_{2n+2}}{a_{2n}}=\frac{1}{2} \text{ and }\frac{a_{2n+3}}{a_{2n+1}}=\frac{1}{3} $$ Hence the radius of convergence is equal to min$(R_1,R_2)=2$.