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

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    $\begingroup$ The series converges for $|x|<2$. $\endgroup$ – Mark Viola May 7 '18 at 16:39

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$(R_1,R_2)=2$.

  • $\begingroup$ thanks a lots @Atmos $\endgroup$ – jasmine May 8 '18 at 2:48

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