Determine the boundedness and monotonicity of the sequence $a_n$ as for $\ln\frac{2n}{n+1}$

Question

I know asking open-ended questions is frowned on this website, but really I don't know where to start

I know what montoncity is increasing, nondecreasing, decreasing, and nonincreasing for all $n$ is called monotonic.

Looking at a graph it doesn't look like:

$$\ln\frac{2n}{n+1}$$

would be monotonic. How can I demonstrate this mathematically?

Same with boundedness. Intuitively I imagine it is unbounded since the numerator will grow faster than the denominator

Answer 1

Try to examine the sequence $b_n := \frac{2n}{n + 1}$ first. Is it monotonic? Is it bounded? What is its limit $L$?

Could you deduce what the limit of $a_n = \ln(b_n)$ is (if it has a limit) in function of $L$? Use the continuity of $\ln$.

If a sequence has a limit, what does that say about its boundedness?

Note that $x < y \Rightarrow \ln(x) < \ln(y)$. How does that help to determine the monotonicity of $a_n$?