Radius of convergence. Picture will give better detail

Question

anonymous · Answer

attachment

anonymous · Answer

I'm unsure if i cancelled everything out as I'm unsure where to go from here

anonymous · Answer

I'm assuming sqrt(n) divided by sqrt(n+1) is just 1? but im unsure how exactly

anonymous · Answer

What test are you using? You should be more clear with your work.

anonymous · Answer

ratio test

anonymous · Answer

I thought so, please do it again.

anonymous · Answer

Is it not right as is?

anonymous · Answer

I did not check, but you should do it right, otherwise it's just a mess.

anonymous · Answer

Im pretty sure its correct. What i am left with is (x*sqrt(n))/(sqrt(n+1)*2)

jtvatsim · Answer

The ratio test requires you to take the limit as n goes to infinity

anonymous · Answer

i know, i just havent been writing the limit sign

rational · Answer

saving paper or ink ;p

anonymous · Answer

essentially all i need to know is that sqrtn divided by sqrt(n+1) is equal to 1

jtvatsim · Answer

essentially... yes

jtvatsim · Answer

so what does that leave you with if that part of the expression is 1?

anonymous · Answer

Ok can i get the proof of that? my thought is to just divide each one by sqrt(n)

jtvatsim · Answer

Sure that works

anonymous · Answer

well then its x/2 which means the radius of convergence is 2

anonymous · Answer

ok. I just wanted to make sure my thought process on the sqrt(n) part was correct

jtvatsim · Answer

Yep, your proposed proof should work as well.

anonymous · Answer

Thank you

jtvatsim · Answer

No problem! :)