The limit comparison test

1. $\sin\frac1n\sim\frac1n$ (since $\frac{\sin x}x\to1$). The harmonic series diverges: diverges.

2. $1-\cos x\sim\frac{x^2}2$ at $0$, so $1-\cos\frac1n\sim\frac1{2n^2}$: converges.

3. $\ln(1+x)\sim x$, so $\ln\left(1+\frac1{n^2}\right)\sim\frac1{n^2}$: converges.

4. $\sqrt{n+1}-\sqrt n=\dfrac1{\sqrt{n+1}+\sqrt n}\sim\dfrac1{2\sqrt n}$, and $\sum\frac1{\sqrt n}$ diverges ($p=\frac12$): diverges. (Alternatively, the partial sums telescope to $\sqrt{N+1}-1\to\infty$.)

Remark. Equivalents are the most efficient tool for positive series. They work only for positive terms (or terms of constant sign): for alternating series an equivalent is not enough.