Cauchy series in Euler number

Calculus Level 3

\[ \large \displaystyle \lim_{n\rightarrow\infty} \sqrt{n} \left( \left( 1+\dfrac{1}{n+1} \right)^{n+1} - \left( 1+\dfrac{1}{n} \right)^{n} \right) = \, ? \]