Series Convergence Divergence: sum[n = 1, infty] [(n! ln(n)) / (n (n + 2)!)]

[BlueSourBalls]

Determine if the following series converges or diverges (question 7).

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\(\displaystyle \displaystyle 7.\quad \sum_{n=1}^{\infty}\, \dfrac{n!\, \ln(n)}{n\, (n\, +\, 2)!}\)

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Here's what I have so far (I am using the ratio test):

\(\displaystyle \displaystyle \lim_{n \rightarrow \infty}\, \left(\dfrac{a_{n+1}}{a_n}\right)\)

. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left(\dfrac{\left[\dfrac{(n\, +\, 1)!\, \ln(n\, +\, 1)}{(n\, +\, 1)\, (n\, +\, 3)!}\right]}{\left[\dfrac{n!\, \ln(n)}{n\, (n\, +\, 2)!}\right]}\right)\)

. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left(\dfrac{(n\, +\, 1)!\, \ln(n\, +\, 1)}{(n\, +\, 1)\, (n\, +\, 3)!}\, \cdot\, \dfrac{n\, (n\, +\, 2)!}{n!\, \ln(n)}\right)\)

. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left(\dfrac{(n\, +\, 1)!\, n\, (n\, +\, 2)!}{(n\, +\, 1)\, (n\, +\, 3)!\, n!}\right)\, \cdot\, \lim_{n \rightarrow \infty}\, \left(\dfrac{\ln(n\, +\, 1)}{\ln(n)}\right)\)

I am not sure if I took a wrong turn, but I am expecting the left limit to equal 0 to eliminate the infinity / infinity. However, I do not know how to deal with the factorials within the limit.

 

[tkhunny]

Determine if the following series converges or diverges (question 7).

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\(\displaystyle \displaystyle 7.\quad \sum_{n=1}^{\infty}\, \dfrac{n!\, \ln(n)}{n\, (n\, +\, 2)!}\)

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Here's what I have so far (I am using the ratio test):

\(\displaystyle \displaystyle \lim_{n \rightarrow \infty}\, \left(\dfrac{a_{n+1}}{a_n}\right)\)

. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left(\dfrac{\left[\dfrac{(n\, +\, 1)!\, \ln(n\, +\, 1)}{(n\, +\, 1)\, (n\, +\, 3)!}\right]}{\left[\dfrac{n!\, \ln(n)}{n\, (n\, +\, 2)!}\right]}\right)\)

. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left(\dfrac{(n\, +\, 1)!\, \ln(n\, +\, 1)}{(n\, +\, 1)\, (n\, +\, 3)!}\, \cdot\, \dfrac{n\, (n\, +\, 2)!}{n!\, \ln(n)}\right)\)

. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left(\dfrac{(n\, +\, 1)!\, n\, (n\, +\, 2)!}{(n\, +\, 1)\, (n\, +\, 3)!\, n!}\right)\, \cdot\, \lim_{n \rightarrow \infty}\, \left(\dfrac{\ln(n\, +\, 1)}{\ln(n)}\right)\)

I am not sure if I took a wrong turn, but I am expecting the left limit to equal 0 to eliminate the infinity / infinity. However, I do not know how to deal with the factorials within the limit.

For starters, \(\displaystyle \dfrac{n!}{(n+2)!} = \dfrac{1}{(n+2)(n+1)}\). Why not simplify your life a little?

Second, the ratio test is inconclusive. What's your next option? Perhaps you could observe that \(\displaystyle ln(n) < n;for\;n\ge 1\)?

 

[BlueSourBalls]

For starters, \(\displaystyle \dfrac{n!}{(n+2)!} = \dfrac{1}{(n+2)(n+1)}\). Why not simplify your life a little?

Second, the ratio test is inconclusive. What's your next option? Perhaps you could observe that \(\displaystyle ln(n) < n;for\;n\ge 1\)?

Wow that does simply my life. I did not see that. Thank you, I think I got it from here.

 

[Steven G]

Determine if the following series converges or diverges (question 7).

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\(\displaystyle \displaystyle 7.\quad \sum_{n=1}^{\infty}\, \dfrac{n!\, \ln(n)}{n\, (n\, +\, 2)!}\)

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Here's what I have so far (I am using the ratio test):

\(\displaystyle \displaystyle \lim_{n \rightarrow \infty}\, \left(\dfrac{a_{n+1}}{a_n}\right)\)

. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left(\dfrac{\left[\dfrac{(n\, +\, 1)!\, \ln(n\, +\, 1)}{(n\, +\, 1)\, (n\, +\, 3)!}\right]}{\left[\dfrac{n!\, \ln(n)}{n\, (n\, +\, 2)!}\right]}\right)\)

. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left(\dfrac{(n\, +\, 1)!\, \ln(n\, +\, 1)}{(n\, +\, 1)\, (n\, +\, 3)!}\, \cdot\, \dfrac{n\, (n\, +\, 2)!}{n!\, \ln(n)}\right)\)

. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left(\dfrac{(n\, +\, 1)!\, n\, (n\, +\, 2)!}{(n\, +\, 1)\, (n\, +\, 3)!\, n!}\right)\, \cdot\, \lim_{n \rightarrow \infty}\, \left(\dfrac{\ln(n\, +\, 1)}{\ln(n)}\right)\)

I am not sure if I took a wrong turn, but I am expecting the left limit to equal 0 to eliminate the infinity / infinity. However, I do not know how to deal with the factorials within the limit.

Why not use that (n+1)*n! = (n+1)! ? How about (n+2)!/(n+3)! = 1/(n+3). Maybe a little L'Hopital rule here and there.

 

[tkhunny]

Maybe a little L'Hopital rule here and there.

Not really a fan of the overuse of that rule. Why not just use a simple comparison test? That overused rule does fail.