| Series | Convergent | Divergent |
|---|---|---|
| The geometric series \(\sum\limits_{k\,=\,1}^{\infty}ar^{k-1}\) | \(\vert ~r \vert <1\) | \(\vert ~r \vert \geq 1\) |
| The harmonic series \(\sum\limits_{k\,=\,1}^{\infty}\dfrac{1}{k}\) | Divergent | |
| The \(p\)-series \(\sum\limits_{k\,=\,1}^{\infty}\dfrac{1}{k^{p}}\) | \(p>1\) | \(0<p\leq 1\) |
| The series \(\sum\limits_{k\,=\,1}^{\infty}\dfrac{1}{k^{k}}\) | Convergent |