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 \(2\sqrt{n}-1\).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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