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#