Bounds on $S = \frac{1}{1001} + \frac{1}{1002}+ \frac{1}{1003}+\dots+\frac{1}{3001}$ 
$S =  \frac{1}{1001} +   \frac{1}{1002}+ \frac{1}{1003}+ \dots+\frac{1}{3001}$.


Prove that $\dfrac{29}{27}<S<\dfrac{7}{6}$.

My Attempt:
$S<\dfrac{500}{1000} +   \dfrac{500}{1500}+ \dfrac{500}{2000}+ \dfrac{500}{2500}+\dfrac{1}{3000} =\dfrac{3851}{3000}$
(Taking 250 terms together involves many fractions and is difficult to calculate by hand.)
Using AM-HM inequality gave me $S > 1$, but the bounds are weak.
Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\dots+\frac{1}{3001}<\frac43$ 
Inequality with sum of inverses of consecutive numbers 
The answers to these questions are nice, but the bounds are weak.
Any help without calculus and without calculations involving calculators would be appreciated.
(I encountered this question when I was preparing for a contest which neither allows calculators nor calculus(Only high-school mathematics.))
 A: We can find an approximation of the result.
Consider
$$\sum_{i=1}^{2001}\frac1{1000+i}=\sum_{i=1}^{3001}\frac1{i}-\sum_{i=1}^{1000}\frac1{i}$$ and we shall use the fact that
$$\sum_{i=1}^{n}\frac1{i}=H_n$$
Now, the asymptotics of harmonic numbers
$$H_n=\gamma +\log \left({n}\right)+\frac{1}{2
   n}-\frac{1}{12 n^2}+O\left(\frac{1}{n^4}\right)$$ Using it twice and you will have sharper bounds.
A: Well not an answer but an idea to inspire someone :
I remenber the Gauss's solution of the first case concerning the Faulhaber's formula :
We have :
$$\frac{1}{1001}+\frac{1}{3001}>\frac{1}{1002}+\frac{1}{3000}>\cdots>\frac{1}{2002}+\frac{1}{2000}$$
Wich gives :
$$S<\frac{1000}{1001}+\frac{1000}{3001}+\frac{1}{2001}$$
Now if we divide (3001-1001) by four we get :
$$\frac{500}{2500}+\frac{500}{2502}+\frac{1}{2501}+\frac{499}{1499}+\frac{499}{1501}+\frac{1}{1500}<S<\frac{500}{1001}+\frac{500}{2000}+\frac{1}{1500}+\frac{499}{2001}+\frac{499}{3001}+\frac{1}{2500}$$
A: $$\sum_{n=1}^{2001}\frac{1}{1000\left(1+\frac{n}{1000}\right)} =\int_{0}^{2}\frac{1}{\left(1+x\right)}dx \approx \ln3$$
$$\frac{29}{27} < \ln3 <\frac{7}{6}$$
Used:

*

*Summation to integration when step size is small

