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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 $p=3$ .
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 $p=\dfrac{1}{2}$ .