What is the radius of convergence of $sum_1^oo (x-1)^n / 2^(2n)?$ ?

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Answer 1 · 2016-07-09T10:19:23

$3 < x < 5$

$sum_1^oo (x-1)^n / 2^(2n)=sum_1^oo ((x-1)/4)^n$

Making $y = (x-1)/4$

$sum_1^oo (x-1)^n / 2^(2n) equiv sum_1^oo y^n$

We know that for $abs y < 1$

$sum_1^oo y^n = 1/(1-y)$

or equivalently

$sum_1^oo (x-1)^n / 2^(2n)=4/(5-x)$ for $abs((x-1)/4)<1$ or $3 < x < 5$

What is the radius of convergence of sum_1^oo (x-1)^n / 2^(2n)?sum_1^oo (x-1)^n / 2^(2n)??