Analyzing a \(p\)-Series

1. The series \[ \sum_{k\,=\,1}^{\infty }\frac{1}{k^{3}}=1+\frac{1}{2^{3}}+\frac{1}{3^{3}} +\cdots +\frac{1}{n^{3}}+\cdots \] converges, since it is a \(p\)-series where \(p=3\).
2. The series \[ \sum_{k\,=\,1}^{\infty }\frac{1}{\sqrt{k}}=1+\frac{1}{\sqrt{2}}+\frac{1}{ \sqrt{3}}+\cdots +\frac{1}{\sqrt{n}}+\cdots \] diverges, since it is a \(p\)-series where \(p=\dfrac{1}{2}\).