anonymous
  • anonymous
Say if this sequence converges or diverges. If converges state limit.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
a= \[4^n/n^4\]
anonymous
  • anonymous
im really not sure how to approach this. pretty sure I take the limit?
anonymous
  • anonymous
yes, you want the limit as n goes to infinity.

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anonymous
  • anonymous
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
anonymous
  • anonymous
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.
anonymous
  • anonymous
How exactly do I show that though?
anonymous
  • anonymous
do you know L'Hopital's rule?
anonymous
  • anonymous
Yes. take the derviative of f(x) over g(x)?
anonymous
  • anonymous
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
anonymous
  • anonymous
I got the derviative for n^4..but struggling with 4^n.. sorry havent done this in ages, cant figure it out
anonymous
  • anonymous
ln(4)4^n?
anonymous
  • anonymous
\[\frac{d}{dx}\left(a^x\right)=a^x \ln(a)\]i believe.
anonymous
  • anonymous
yeah you got it lol
anonymous
  • anonymous
So dont I use the rule agian since that doesnt help me? still infinity over infinity..
anonymous
  • anonymous
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*
anonymous
  • anonymous
so the top is going to stay infinty, bottom 0, so its going to be infinity?
anonymous
  • anonymous
yep, thats the gist of it :)
anonymous
  • anonymous
ty

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