I have $$M_{n}=\sup_{x\in\left[1,2\right]}\left|\frac{x}{\left(x+1\right)^{n}}\right|$$
How do I evaluate this? Am I supposed to fix $n\in\mathbb{N}$ before trying to find the supremum? I tried doing that, but I see that if $n=1$, then the supremum occurs at $x=2$, but if $n\neq1$, then the supremum occurs at $x=1$. Or, am I not supposed to fix $n$ at all?
You are supposed to fix $n$ before taking the supremum over $x \in [1,2]$.
Yes, the supremum occurs at $x = 2$ if $n=1$, but at $x = 1$ if $n=2$. However, if you plot the graphs, you'll discover that the supremum actually occurs at $x = 1$ for all $n \geq 2$. (Presumably you can also prove this rigorously by standard techniques, e.g. by differentiating...)
Since your ultimate goal is to apply the Weierstrass M-test, it doesn't matter if you throw away a single term. So you may as well compute $M_n$ for $n \geq 2$ only, and then try to determine whether or not $\sum_{n=2}^\infty M_n$ is a convergent series.