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How to calculate the integer part of the value of the following equation?

$$y=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\ldots+\frac{1}{\sqrt{1000000}}$$

It should be calculated in a special way, after all the equation is so long.

Julien
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Railgun
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1 Answers1

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We have from the decreasing of the function $x\mapsto\frac{1}{\sqrt x}$ $$\sum_{k=1}^n\int_{k}^{k+1}\frac{dx}{\sqrt x}\le\sum_{k=1}^n\frac{1}{\sqrt k}\leq\sum_{k=2}^n\int_{k-1}^k\frac{dx}{\sqrt x}+1$$ hence

$$ 2(\sqrt{n+1}-1)=\int_1^{n+1}\frac{dx}{\sqrt x}\le\sum_{k=1}^n\frac{1}{\sqrt k}\leq\int_{1}^n\frac{dx}{\sqrt x}+1= 2(\sqrt n-1)+1$$

Numerically for $n=1000000$ the integer part is $1998$

  • Nice approach, but shouldn't the result be $1 + 1997$ from http://www.wolframalpha.com/input/?i=sum+1%2Fsqrt%28n%29%2C+%28n%2C2%2C1000000%29? – Amzoti Oct 26 '13 at 16:12
  • Great!But I wonder whether it can be solved by more elementary method. – Railgun Oct 26 '13 at 16:41