Question

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

Answer 1

32

Explanation:

`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