Can the ratio test imply convergence to zero for a sequence?
for example show n! / n^n converges,

- anonymous

Can the ratio test imply convergence to zero for a sequence?
for example show n! / n^n converges,

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- anonymous

ahh ratio test is useless here as well

- anonymous

double whammy

- 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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## More answers

- anonymous

the limit produces 1^n

- anonymous

1 ^ infinity, so indeterminate?

- anonymous

ahh, there is still hope

- 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

wow

- 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

now using sandwhich theorem...

- anonymous

we know that lim r^n for | r | < 1 is 0 , right?

- 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

right

- anonymous

whats an example of a non archimedian ordered field?

- anonymous

I don't know.

- anonymous

youre a math genius !!! . to get that epsilon proof thanks

- anonymous

i am humbled by your solution

- 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

This is very cool. watch this

- 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

so we can use this as a lemma in our sequence n! / n^n

- anonymous

lim n→+∞ (1 - 1/n)ⁿ = 1/e

- anonymous

n!/n^n<(1/n)*(1-1/n)^(n-2)= [n/(n-1)^2] * (1 – 1/n)^n = 0 * 1/e

- anonymous

yep, both pretty slick solutions

- 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

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

0< n!/n^n < 0 in the limit as n goes to infinity
i bet there is a sandwich theorem for sequences

- anonymous

interesting we have
lim n→+∞ (1 - 1/n)ⁿ = 1/e
and
lim n→+∞ (1 + 1/n)ⁿ = e

- anonymous

watch out, this website will freeze your work
i type it in notepad first

- 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

0 * e^-1

- 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

nope that wont work

- 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

the series converges , not necessarily the sequence. Im trying to think of a counterexample...

- 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

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

nope, the original question was for the sequence n! / n^n, show that it converges to zero

- anonymous

....the author doesn't know how to put a sigma sign....It's not that complicated of a question. Trust me

- anonymous

right, the ratio test shows that a positive sequence is decreasing. because of the condition L < 1

- 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

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

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

Yes that works. Since the series converges, then limit must be 0

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