What is the radius of convergence of $sum_1^oo ((2x)^n ) / 8^(2n)$ ?

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Answer 1 · 2016-03-08T18:31:58

32

$sum_1^oo ((2x)^n ) / 8^(2n)$ = $sum_1^oo ((2x) / 8^(2))^n$ = $sum_1^oo ((x) / 32)^n$

We can stop here using the radius definition, but for completeness:

$sum_1^oo ((x) / 32)^n = lim_(n->oo) (1-(x/32)^n)/(1-x/32)$

Which converges only if $|x/32| < 1 => |x| < 32$

Therefore the ratio of convergence is 32

What is the radius of convergence of sum_1^oo ((2x)^n ) / 8^(2n)sum_1^oo ((2x)^n ) / 8^(2n)?