anonymous
  • anonymous
Can the ratio test imply convergence to zero for a sequence? for example show n! / n^n converges,
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
ahh ratio test is useless here as well
anonymous
  • anonymous
double whammy
anonymous
  • anonymous
you get (n+1)! / (n+1)^ n+1 * n^n / n! = n^n / (n+1)^n = (n / n+1)^n

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anonymous
  • anonymous
the limit produces 1^n
anonymous
  • anonymous
1 ^ infinity, so indeterminate?
anonymous
  • anonymous
ahh, there is still hope
anonymous
  • anonymous
You can try the definition of convergence. We want to show that n!/n^n converges to 0. So we’d like to show that for any epsilon > 0, we can find some positive integer M such that for all positive integers N > M, we’d have that N!/N^N is less than epsilon. This would show that n!/n^n converges to 0. Proof: Let epsilon be some arbitrarily small positive number less than 1/2. We know that if we wanted to, we could find some positive integer J such that 0 < 1/J < epsilon, since the sequence {1/n} converges to 0. We wanted to show that we can find some positive integer M such that for all positive integers N > M, we’d have that N!/N^N is less than epsilon. Instead, we will show that we can find some positive integer M such that for all positive integers N > M, we’d have that N!/N^N is less than 1/J. This would imply that N!/N^N < epsilon. My claim is that you can let M be J, because J!/J^J is less than 1/J for all J > 2 (which holds since epsilon is less than 1/2). We can show that J!/J^J < 1/J by showing that J! < J^(J-1), which we can do by induction (omitted). Therefore, for any epsilon > 0, we can find some positive integer M such that for all positive integers N > M, we’d have that N!/N^N is less than epsilon. So n!/n^n converges to 0.
anonymous
  • anonymous
wow
anonymous
  • anonymous
what about this idea n!/n^n=1*2*..*(n-1)*n/(n*n*...*n)= =(1/n)*{(2/n)*...*(1-2/n)}*(1-1/n)*1 Each term in braces is less than (1-1/n). There are (n-2) terms there, so n!/n^n<(1/n)*(1-1/n)^(n-2)*1
anonymous
  • anonymous
now using sandwhich theorem...
anonymous
  • anonymous
we know that lim r^n for | r | < 1 is 0 , right?
anonymous
  • anonymous
Yeah I think that n!/n^n<(1/n)*(1-1/n)^(n-2)*1 also works. You'd be able to rewrite the right side as [n/(n-1)^2] * (1 – 1/n)^n, which converges to 0.
anonymous
  • anonymous
right
anonymous
  • anonymous
whats an example of a non archimedian ordered field?
anonymous
  • anonymous
I don't know.
anonymous
  • anonymous
youre a math genius !!! . to get that epsilon proof thanks
anonymous
  • anonymous
i am humbled by your solution
anonymous
  • anonymous
but wait, there is more i can do. but just wanted to ask, the ratio test doesnt help us here, but if the ratio test converges to zero, does that mean that the sequence must converge to zero ( i know the series will converge)
anonymous
  • anonymous
This is very cool. watch this
anonymous
  • anonymous
lim n→+∞ [(1 - 1/n)⁽⁻ⁿ⁾]⁻¹ = lim n→+∞ {[(n - 1) /n ]⁽⁻ⁿ⁾}⁻¹ = lim n→+∞ {{1/ [(n - 1) / n ]} ⁿ}⁻¹ = lim n→+∞ {[ n /(n - 1)]ⁿ}⁻¹ = you can easily verify that n / (n - 1) can be written as 1+1/(n - 1), so: lim n→+∞ {[ n /(n - 1)]ⁿ}⁻¹ = lim n→+∞ {[ 1+1/(n -1)]ⁿ}⁻¹ = lim n→+∞ {[1+1/(n -1)]∙[1+1/(n -1)]⁽ⁿ⁻¹⁾}⁻¹ = notice that if n→+∞ then (n -1) →+∞ too, therefore lim n→+∞ [1+1/(n -1)]⁽ⁿ⁻¹⁾→ e and, finally lim n→+∞ [(1+0)∙e]⁻¹ = 1/e
anonymous
  • anonymous
so we can use this as a lemma in our sequence n! / n^n
anonymous
  • anonymous
lim n→+∞ (1 - 1/n)ⁿ = 1/e
anonymous
  • anonymous
n!/n^n<(1/n)*(1-1/n)^(n-2)= [n/(n-1)^2] * (1 – 1/n)^n = 0 * 1/e
anonymous
  • anonymous
yep, both pretty slick solutions
anonymous
  • anonymous
actually we need that lemma, right , i think. or .. since it is not lim r^n for |r|< 1 , not quite the same thing
anonymous
  • anonymous
and then here i used sandwich, 0< n!/n^n<(1/n)*(1-1/n)^(n-2)= [n/(n-1)^2] * (1 – 1/n)^n = 0 * 1/e hmmm, can you sandwich sequences, i dont know
anonymous
  • anonymous
0< n!/n^n < 0 in the limit as n goes to infinity i bet there is a sandwich theorem for sequences
anonymous
  • anonymous
interesting we have lim n→+∞ (1 - 1/n)ⁿ = 1/e and lim n→+∞ (1 + 1/n)ⁿ = e
anonymous
  • anonymous
watch out, this website will freeze your work i type it in notepad first
anonymous
  • anonymous
you have to be careful to keep the limits. otherwise, you get nonsense results like 0 < 0. we showed that n!/n^n < [n/(n-1)^2] * (1 – 1/n)^n So taking the limit as n goes to infinity of both sides gives lim n!/n^n <= lim [n/(n-1)^2] * (1 – 1/n)^n But the right side equal lim [n/(n-1)^2] * lim (1 – 1/n)^n = 0 * e = 0. Since values of n are all positive, it follows that lim n!/n^n = 0.
anonymous
  • anonymous
0 * e^-1
anonymous
  • anonymous
....you do realize it's getting complicated here for someone that is probably not at the level of using epsilon/N definition in their HW. Just use the ratio test and you get (n/n+1)^n like you said. So then... \[(n/n+1)^n = (1/(1+1/n))^n\] We see that this approaches 1/e which is less than 1. So this series converges
anonymous
  • anonymous
nope that wont work
anonymous
  • anonymous
Well, the author's screen name was cantorset, so I assumed he's taking upper division math classes in college. The proof is pretty standard in those classes. The problem is asking us to find what a sequence converges to, not what the sum converges to. Ratio tests for sequences can show that a sequence is decreasing, but not necessarily what it's decreasing to.
anonymous
  • anonymous
the series converges , not necessarily the sequence. Im trying to think of a counterexample...
anonymous
  • anonymous
right, we wanted to show that the sequence converges to a particular value , here zero. so the ratio test is of no luck here
anonymous
  • anonymous
No, the ratio test is simply to determine if the series is convergent. The original question is asking for if the series do infact converges. Not the series.
anonymous
  • anonymous
nope, the original question was for the sequence n! / n^n, show that it converges to zero
anonymous
  • anonymous
....the author doesn't know how to put a sigma sign....It's not that complicated of a question. Trust me
anonymous
  • anonymous
right, the ratio test shows that a positive sequence is decreasing. because of the condition L < 1
anonymous
  • anonymous
It's one of the fundamental mistakes people make to be confused between sequence and series. Cause often they treat it like the same thing.
anonymous
  • anonymous
Given that the sequence is positive, decreasing, and bounded below by zero. the series must converge to some value . but by the divergence test since the seris converges then the lim a_n must go to zero. if lim a_n does not to go zero then the series diverges. so the contrapositive i think solves this problem
anonymous
  • anonymous
well we need to prove the ratio test here. but i think we can use the divergence test (the contrapositive) because the series n! / n^n converges by ratio test, then the limit of the sequence must go to zero . remember divergence test?
anonymous
  • anonymous
Yes that works. Since the series converges, then limit must be 0

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