Is there a theorem relating sequences to its series or vice versa? I only have these in mind

Theorem:
If a series $\sum_n a_n$ of real numbers converges then $\lim_\limits{n \to \infty} |a_n|=0$

Divergence test:
If $\lim_\limits{n \to \infty} a_n \neq 0$, then the series  $\sum_n a_n$ diverges.

are there stronger/other statements?
There is the "ratio test":  The series $\sum a_n$ converges if  the limit of $\frac{a_{n+1}}{a_n}$ is less than 1 and diverges if it is larger than 1.

Or the "root test": the series $\sum a_n$ converges if the limit of $\sqrt[n]{a_n}$ is less than 1.
There is the "ratio test":  The series $\sum a_n$ converges if  the limit of $\frac{a_{n+1}}{a_n}$ is less than 1 and diverges if it is larger than 1.

Or the "root test": the series $\sum a_n$ converges if the limit of $\sqrt[n]{a_n}$ is less than 1.