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?
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4$\begingroup$ Hint: You're vastly overkilling it. $\endgroup$– Git GudJun 6, 2014 at 11:39
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$\begingroup$ Why proving that a finite sum is in $(0.5,1)$? Just calculate the value and you're done. $\endgroup$– ThekwastiJun 6, 2014 at 11:42
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$\begingroup$ similar math.stackexchange.com/q/688432/129458 $\endgroup$– OBDAJun 6, 2014 at 11:56
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1 Answer
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$$\frac{1}{2}=\frac{1000}{2000}<\underbrace{\frac{1}{1001}+\cdots+\frac{1}{2000}}_{1000 \text{ terms}}<\frac{1000}{1000}=1$$
