Convergence of a logarithmic sequence

Question

Is the sequence $\left\{\ln\left((1+\frac1n)^n\right)\right\}_{n=1}^{\infty}$ convergent or divergent?.

I tried to solve it by L Hospital's rule and arrived at 0...implying it is convergent..is it? If it is right then is there an alternate method?

Answer 1

HINT

Recall that by standard limit

$$\left(1+\frac1n\right)^n\to e$$

As an alternative we have by $x=\frac 1n \to 0$

$$\ln \left(1+x\right)^{\frac1x}=\frac{\ln (1+x)}{x}$$

which is also a standard limit and which, as an alternative, can be solved by l'Hopital rule.