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

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Answer 1 · 2016-11-06T09:56:09

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

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 $1/3 <= x <=1$

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