Question

Say if this sequence converges or diverges. If converges state limit.

Answer 1

a= \[4^n/n^4\]

Answer 2

im really not sure how to approach this. pretty sure I take the limit?

Answer 3

yes, you want the limit as n goes to infinity.

Answer 4

How do I find this limit? I honestly havent done theese in ages so not sure.. Im pretty sure its infinity just not sure how to say it

Answer 5

you are correct, the limit doesnt converge. its because exponential functions ( like 4^x) grow way faster than polynomials. The numerator will grow faster than the denominator.

Answer 6

How exactly do I show that though?

Answer 7

do you know L'Hopital's rule?

Answer 8

Yes. take the derviative of f(x) over g(x)?

Answer 9

yes :) since if you try to take the limit straight you get:\[\frac{\infty}{\infty}\] you can use L'Hopital's rule. Take the derivative of both f and g

Answer 10

I got the derviative for n^4..but struggling with 4^n.. sorry havent done this in ages, cant figure it out

Answer 11

ln(4)4^n?

Answer 12

\[\frac{d}{dx}\left(a^x\right)=a^x \ln(a)\]i believe.

Answer 13

yeah you got it lol

Answer 14

So dont I use the rule agian since that doesnt help me? still infinity over infinity..

Answer 15

yep, gotta keep going. notice though that instead of a 4 degree polynomial in the denominator, you have a 3rd degree, and that will keep getting smaller until...*poof*

Answer 16

so the top is going to stay infinty, bottom 0, so its going to be infinity?

Answer 17

yep, thats the gist of it :)

Answer 18

ty