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We need to find the radius of convergence of $\sum_{i=1}^n a_n$ where $a_n$=$(-4)^n*(x+2)^(2n)$/$n(n+1)$ . By ratio test the series converges for |x+2|<1/4 i.e. -9/4<x<-7/4 then what would be the radius of convergence? I know that if series converges for x=a then radius of convergence is R=0 but is it 1/4 here?

Alolika Ray
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  • Yes, that $1/4$ is the radius of convergence. A "radius" is the distance from the center to the edge (that's how it works for circles); so for an interval, the radius is just half the interval length. – Greg Martin Nov 07 '20 at 01:41
  • Yeah Thank you!Greg Martin – Alolika Ray Nov 07 '20 at 01:49

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By Cauchy-Hadamard, the radius of convergence is: $r=\limsup_{n\to\infty}\dfrac1{\sqrt[2n]{\dfrac{4^n}{n(n+1)}}}=\dfrac12$.