Question

What is the interval of convergence of `sum (3x-2)^(n)/(1+n^(2))`?

Answer 1

The interval of convergence is `1/3 <= x <=1`

Explanation:

Let's do the ratio test

`L=lim∣(3x-2)^(n+1)/(1+(n+1)^2)*(1+n^2)/(3x-2)^n∣`
`color(white)(aaaa)` `n->oo`

`=lim∣(3x-2)/(1+(n+1)^2)*(1+n^2)∣`
`color(white)(aaaa)` `n->oo`

`=(3x-2)lim∣(1+n^2)/(1+(n+1)^2)∣`
`color(white)(aaaaaaaaaa)` `n->oo`

The series converge when `∣3x-2∣<1`
So `3x-2<1` `=>` `3x<3` `=>` `x<1`
and `-3x+2<1` `=>` `x>1/3`

So the interval of convergence is `1/3 <= x <=1`