Question

How do you find the radius of convergence of a power series?

Answer 1

I used to live in Hicksville too, when I was a kid!

To find the radius R of convergence of a power series
`sum_(n=0)^(oo)c_n (x-a)^n,` centered at `x=a`, use the Ratio Test,
and check that `lim_(n->oo) |(c_(n+1) (x-a)^(n+1))/(c_n (x-a)^n)|<1,` the same as
`lim_(n->oo) |(c_(n+1))/(c_n)|*|x-a|<1,` or
`|x-a|< lim_(n->oo) |(c_n)/(c_(n+1))|`

We wanted to find R such that our power series converged for
`a-R < x < a+R`, which is `|x-a| < R`, so we use the value
`R = lim_(n->oo) |(c_n)/(c_(n+1))|` for the radius of convergence.

Note: The word "radius" comes from the ability to use complex numbers for our variable x (and also the coefficients), and saying `|x-a| < R` describes the inside of a circle of real radius R in the complex plane, centered at the complex number a.

| dansmath strikes again |