If $\alpha$ is an element of $]0,1[$, show, for example by using a series-integral comparison, that
$$\sum _ { k = 1 } ^ { n } \frac { 1 } { k ^ { \alpha } } \underset { n \rightarrow + \infty } { \sim } \frac { n ^ { 1 - \alpha } } { 1 - \alpha }$$
If $\alpha$ is an element of $]1 , + \infty[$, show similarly that
$$\sum _ { k = n + 1 } ^ { + \infty } \frac { 1 } { k ^ { \alpha } } \underset { n \rightarrow + \infty } { \sim } \frac { 1 } { ( \alpha - 1 ) n ^ { \alpha - 1 } }$$