radius of convergence and the interval of convergence for the given power series

Question

math2400 · Answer

$$\sum_{n=1}^{\infty} \frac{ n ^{2}x ^{2n} }{ n! }$$

math2400 · Answer

my radius is 1. And it think the interval is (-1,1)
Can anyone verify if that's right?

zarkon · Answer

what test did you use?

math2400 · Answer

ratio

zarkon · Answer

do it again

math2400 · Answer

ok

math2400 · Answer

wait can the radius be infinity??

zarkon · Answer

yes

math2400 · Answer

ok..and the interval is (-infin, infin) ?

zarkon · Answer

yes

math2400 · Answer

lol kk thanks. that's what i got first but i wasn't sure if that was possible haha

xapproachesinfinity · Answer

i got oo as well :)

math2400 · Answer

perfect thanks!! :)

xapproachesinfinity · Answer

limit (x^2 . (n+1/n)^2/(n+1))=0

math2400 · Answer

how about this one if u don't mind: $$\sum_{n=1}^{\infty} \frac{ n^{e} }{ e ^{n} }$$  i'm pretty sure the radius is 1 right?

xapproachesinfinity · Answer

that says for any value of x rho is always zero

math2400 · Answer

and then interval is (2,4) or maybe (2,4] not sure, i suck at these

xapproachesinfinity · Answer

i would do root test with that one

xapproachesinfinity · Answer

did you try ratio?

zarkon · Answer

there is no x in your last problem

math2400 · Answer

yes(: I got the radius to be 1 and then i know the interval is (2,4) but I'm not sure

xapproachesinfinity · Answer

eh hold on you are asking for convergence because there is not x

xapproachesinfinity · Answer

you need x

math2400 · Answer

ohh wait sorry i posted the wrong one. That one is a converging series haha. I meant to post this one: $$\sum_{n=1}^{\infty} \frac{ (x-3)^n }{ \sqrt{n} }$$

xapproachesinfinity · Answer

oh ok
i would go with root test with that too
limit n^1/n=1
rho=|x-3|<1
so raduis is 1
-1<x-3<1 ===> 2<x<4

xapproachesinfinity · Answer

i guess you got this :)

math2400 · Answer

ok so radius is 1 and then did u check the end points?

xapproachesinfinity · Answer

No i didn't x=4
you have series 1/n^1/2 diverges
x=2
(-1)^n/rootn
i believe this converge by alternating series test

math2400 · Answer

so would it be (2,4] or just (2,4) that's what I'm confused on

xapproachesinfinity · Answer

no converge on 2 not on 4
so [2,4)
otherwise 2<=x<4

math2400 · Answer

ohh haha kk got it :) Thank you so much!

xapproachesinfinity · Answer

no problem :)

xapproachesinfinity · Answer

actually you seem to me good at this, just few more practice and you master it

math2400 · Answer

haha thank you, means a lot @xapproachesinfinity