Greatest integer less than $\sum_{n=1}^{1599} \frac{1}{\sqrt n}$

Asked Aug 21 '20 at 16:53

Active Aug 21 '20 at 16:59

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What is the greatest integer not exceeding the sum $$\sum_{n=1}^{1599} \frac{1}{\sqrt n}$$

I have already arrived at a solution using calculus but since this came in a Math Olympiad which do not include calculus in their syllabus I wanted to know how I could do this without it. My first thought was to use Titu's Lemma but I couldn't think of what I could do after that. Any hints will be appreciated, thanks

asked Aug 21 '20 at 16:53

Amadeus

1,097

This has been asked and answered several times before: https://math.stackexchange.com/q/516846/42969, https://math.stackexchange.com/q/2600796/42969, https://math.stackexchange.com/q/2522888/42969, https://math.stackexchange.com/q/2296145/42969. The upper bound varies, but the method is the same. – Martin R Aug 21 '20 at 17:12
Possible duplicate of $\left\lfloor \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{1024}}\right\rfloor =?$ – Martin R Aug 21 '20 at 17:20

Greatest integer less than $\sum_{n=1}^{1599} \frac{1}{\sqrt n}$

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