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

1 Answer
Narad T.
Nov 6, 2016

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#

Related questions
Impact of this question
2297 views around the world
You can reuse this answer
Creative Commons License