# Upper and lower bounds for series and sequences [closed]

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, NChFeb 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 $$n$$.