Prove that ${1\over2}<{1\over1001}+......+{1\over2000}<1$

Can it be proved by langrange's mean value theorem or by convert it into a Riemann sum?

  • 4
    $\begingroup$ Hint: You're vastly overkilling it. $\endgroup$ – Git Gud Jun 6 '14 at 11:39
  • $\begingroup$ Why proving that a finite sum is in $(0.5,1)$? Just calculate the value and you're done. $\endgroup$ – Thekwasti Jun 6 '14 at 11:42
  • $\begingroup$ similar math.stackexchange.com/q/688432/129458 $\endgroup$ – OBDA Jun 6 '14 at 11:56

$$\frac{1}{2}=\frac{1000}{2000}<\underbrace{\frac{1}{1001}+\cdots+\frac{1}{2000}}_{1000 \text{ terms}}<\frac{1000}{1000}=1$$


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