n! / ((n/e) ^ n)

Question

converges or diverges

 

 

 

 

 

i think if i use ratio test ill end up with 1 which is inconclusive

 

help pls

Answer 1

As stated above, the ratio is e/((n+1)/n)^n

 

Let's focus on the log of the ratio.  This is 1 - n ln(1 + 1/n))

Next is the key step.  Rather than approximating ln(1 + 1/n) with 1/n, approximate it with 1/n - 1/2n^2

Then, 1 - n ln(1 + 1/n)) is approximately 1 - n(1/n - 1/2n^2) = 1/2n, not 0

 

Note that the sum of logs equals the log of the product of the ratios and may be approximated by an integral.

 

Of course, the integral of 1/2n from a to b is  1/2 ln(b) - 1/2 ln(a) = 1/2 ln(b/a)

 

Thus, the ratio of the product at the b'th term to the a'th term is e^(1/2 ln(b/a)) = sqrt(b/a)

 

This term goes to infinity as b goes to infinity, so the ratio diverges.

 

To show this approximation is reasonable, at 100, our term is 25.08717995, and at 1000, our term is 79.27315177.

The ratio is 3.15990685.

sqrt(1000/100) = sqrt(10) = 3.16227766, which is close.

Answer 2

Find whether lim as x goes to infinity of   n!eⁿ/(nⁿ) converges or diverges.

 

Ratio test says examine limit of  (n+1)!e⁽ⁿ⁺¹⁾/(n+1)⁽ⁿ⁺¹⁾ ÷  n!eⁿ/(nⁿ).

 

This ratio can be simplified to e(n+1)[nⁿ/(n+1)nⁿ⁺¹] or further to enⁿ/(n+1)ⁿ or

 

(by multiplying top & bottom by reciprocal of nⁿ)

 

e ÷ [(1+1/n)ⁿ] and when we take the limit of that as x goes to infinity we get e/e = 1.

 

So I agree with Norbert: INCONCLUSIVE, based on the above details.

 

 

Answer 3

Pia,

 

Using the Ratio Test produces the limit

 

lim_n→∞ (n/(n+1))ⁿ 

 

after simplifying.

 

This can be rewritten as

 

lim_n→∞ (1 + 1/n)^-n = [lim_n→∞ (1 + 1/n)ⁿ]^-1 = e^-1 < 1.

 

Since the limit is less than one, the series will converge.

 

I left out a lot of algebraic details, so feel free to reply if you need them.