The comparison test in practice

1. $\frac1{n^2+n}\le\frac1{n^2}$, a convergent $p$-series ($p=2$): converges. (It even telescopes: $\frac1n-\frac1{n+1}$.)

2. $\frac1{\sqrt{n^3+1}}\le\frac1{n^{3/2}}$, a convergent $p$-series ($p=\frac32>1$): converges.

3. For $n\ge2$, $n^2-n+1\le n^2$, so $\frac{n+1}{n^2-n+1}\ge\frac{n+1}{n^2}\ge\frac1n$. The harmonic series diverges: diverges.

4. $0\le\frac{\sin^2n}{2^n}\le\frac1{2^n}$, a convergent geometric series: converges.

Rule of thumb. Look at the dominant behaviour of each term for large $n$ (here $\frac1{n^2}$, $\frac1{n^{3/2}}$, $\frac1n$), then find an inequality in the right direction.