Find upper and lower bounds for the following finite sum: $$1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}}$$


closed as off-topic by TravisJ, GNUSupporter 8964民主女神 地下教會, Thomas Shelby, RRL, NCh Feb 15 at 7:23

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It depends how much do you want to be tight this upper and lower bound. However, As we have $$\frac{1}{\sqrt{n}}+ \cdots + \frac{1}{\sqrt{n}} = \frac{n}{\sqrt{n}} = \sqrt{n} \leq 1+\frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \leq 1 + \cdots + 1 = n$$

A lower bound is $\sqrt{n}$ and an upper bound is $n$.

  • $\begingroup$ are there any details you omitted? Is it how it's done? $\endgroup$ – Lola Feb 11 at 15:50
  • $\begingroup$ @Lola it's updated. there is no more details. $\endgroup$ – OmG Feb 11 at 16:10
  • $\begingroup$ okay thank you for your help $\endgroup$ – Lola Feb 11 at 16:11

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