Where did this step come from? Problem on series converge/divergence.

[Lo.Lee.Ta.]

1. The original problem is this, but I'm just trying to figure out this one particular step!
∞
Ʃ n!/(nⁿ)
n=1

2. I'm at this step:

lim ln(n/n+1)/(1/n) = lnp
n→∞

BUT THEN my teacher wrote out the next step, and I'm like, "What the heck is that?!"

This is the next step:

lim \frac{\frac{1}{n/(n+1)}*\frac{(n+1)*1-n*1}{(n+1)^{2}}}{-1/n^{2}}
n→∞

Aaaaaagh! What the heck is going on?!
How did he get to this step? O_O

I am stuck wondering how I am supposed to figure out the limit at: lim\frac{ln(n/(n+1)}{1/n}!

=_= HELP PLEASE!
Thank you!

 

[drawar]

1. The original problem is this, but I'm just trying to figure out this one particular step!
∞
Ʃ n!/(nⁿ)
n=1

2. I'm at this step:

lim ln(n/n+1)/(1/n) = lnp
n→∞

BUT THEN my teacher wrote out the next step, and I'm like, "What the heck is that?!"

This is the next step:

lim \frac{\frac{1}{n/(n+1)}*\frac{(n+1)*1-n*1}{(n+1)^{2}}}{-1/n^{2}}
n→∞

Aaaaaagh! What the heck is going on?!
How did he get to this step? O_O

I am stuck wondering how I am supposed to figure out the lim it at: lim\frac{ln(n/(n+1)}{1/n}!

=_= HELP PLEASE!
Thank you!

He was applying the L'Hospital's Rule to a limit of indeterminate form 0/∞.

 

[SammyS]

He was applying the L'Hospital's Rule to a limit of indeterminate form 0/∞.

$\displaystyle \lim_{n\to\,\infty} \left( \frac{\displaystyle \ln\left(\frac{n}{n+1}\right)}{\displaystyle \frac{1}{n}} \right)\ \$ is of the indeterminate form, 0/0 .

 

[drawar]

$\displaystyle \lim_{n\to\,\infty} \left( \frac{\displaystyle \ln\left(\frac{n}{n+1}\right)}{\displaystyle \frac{1}{n}} \right)\ \$ is of the indeterminate form, 0/0 .

Oops, sorry, just ignore what I've said.