calculus - Determine $lim_{nto infty} nleft(1+(n+1)ln frac{n}{n+1}right)$

Determine

$$\lim_{n\to \infty} n\left(1+(n+1)\ln \frac{n}{n+1}\right)$$
I noticed the indeterminate case $\infty \cdot 0$ and I tried to get them all under the $\ln$, but it got more complicated and I reached another indeterminate form. The same happened when I tried to use Stolz-Cesaro.

EDIT: is there an elementary solution, without l'Hospital or Taylor series?

Answer

Let $\frac {n}{n+1}=e^x.$ As $n\to \infty,\;\; x\to 0$ through negative values.The expression is

$$e^x\cdot \frac {1+x-e^x}{(1-e^x)^2}.$$ Applying l'Hopital's Rule to $\frac {1+x-e^x}{(1-e^x)^2}$ we get a limit of $\frac {-1}{2}$ as $x\to 0.$ And the far left term $e^x$ in the expression goes to $1$ as $x\to 0.$