Estimation of the sum of the series
Task number: 3855
Estimate the sum of the series \(\sum\limits_{k=1}^n \frac{1}{\sqrt{k}}\).
Solution
Either by the integration, or by using estimates:
\(\frac{1}{\sqrt{k}} \geq \frac{2}{\sqrt{k}+\sqrt{k+1}} = 2(\sqrt{k+1}-\sqrt{k})\) and
\(\frac{1}{\sqrt{k}} \leq \frac{2}{\sqrt{k}+\sqrt{k-1}} = 2(\sqrt{k}-\sqrt{k-1})\)
we get thet \(2(\sqrt{n+1}-1) \leq \sum\limits_{k=1}^n \frac{1}{\sqrt{k}} \leq 2\sqrt{n}\).
The upper bound could be improved on \(2\sqrt{n}-1\).
