Help with proving a sequence is convergent

Question

Prove that the sequence $\{\ln(1+\frac{1}{n})^n\}_{n=1}^\infty$ converges. I don't know how to approach this problem without the use of limits. Any insight would be helpful.

Answer 1

You can prove this using the Monotone Convergence theorem. $\ln(x)$ is monotone increasing (to show this just look at its derivative). Using the binomial theorem one can prove that $(1+\frac{1}{n})^n $ is bounded (by $e$). In fact, $(1+\frac{1}{n})^n $ this is also monotone. Here is a post discussing this: Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing. This completes the proof, since $\ln(1+\frac{1}{n})^n$ is monotone increasing and bounded above by $\ln(e)=1$

Answer 2

If you know that $\left(1+\frac{1}{n}\right)^n$ goes to $e$, as $n$ goes to infinity, you only have to use the continuity of the $\ln$ function.

Answer 3

$lim_{n\rightarrow\infty}\ln(1+1/n)^n=\ln[lim_{n\rightarrow\infty}(1+1/n)^n]=\ln_e e =1.$

As explained in other answers the limit property that applied here follows from the continuity of $\ln x.$