What exactly is this limit question asking me to do? I have a homework question that is as follows:
Calculate

$$\lim _{ n\to \infty  } \left[ \frac { 1 }{ n+1 } +\frac { 1 }{ n+2 } +...+\frac { 1 }{ 2n }  \right] $$

by writing the expression in brackets as
$$\frac { 1 }{ n } \left[ \frac { 1 }{ 1+1/n } +\frac { 1 }{ 1+2/n } +...+\frac { 1 }{ 1+n/n }  \right] $$
and recognizing the latter as a Riemann sum.
I am aware of what a Riemann sum is, but not quite sure what the first expression is depicting the sum of. The second expression makes almost no sense to me and I am not sure what the question is general is trying to get me to do. Any help would be greatly appreciated as I do not directly want the answer, just examples and guiding steps towards being able to solve it myself. Thanks!
 A: $$\lim _{ n\to \infty  } \left[ \frac { 1 }{ n+1 } +\frac { 1 }{ n+2 } +...+\frac { 1 }{ 2n }  \right] =\lim _{ n\to \infty  } \frac { 1 }{ n } \left[ \frac { 1 }{ 1+\frac { 1 }{ n }  } +\frac { 1 }{ 1+\frac { 2 }{ n }  } +...+\frac { 1 }{ 1+\frac { n }{ n }  }  \right] =\\ =\lim _{ n\to \infty  } \frac { 1 }{ n } \sum _{ k=1 }^{ n }{ \frac { 1 }{ 1+\frac { k }{ n }  }  } =\int _{ 0 }^{ 1 }{ \frac { 1 }{ 1+x } dx } =\color{red}{\ln 2}  $$
A: Hint:
Write the second expression as
$$\frac1n\sum_{k=1}^n\frac1{1+\frac kn}=\frac1n\sum_{k=1}^n\frac1{1+x_k}$$
if we set $x_k=\frac kn$.
A: Beside the Riemann sum, you also approach the problem using harmonic numbers 
$$S_n=\sum_{i=1}^n\frac 1{n+i}=H_{2 n}-H_n$$ and using the asymptotics $$H_p=\gamma +\log \left(p\right)+\frac{1}{2 p}-\frac{1}{12
   p^2}+O\left(\frac{1}{p^3}\right)$$ you will get $$S_n=\sum_{i=1}^n\frac 1{n+i}=H_{2 n}-H_n=\log (2)-\frac{1}{4 n}+\frac{1}{16 n^2}+O\left(\frac{1}{n^3}\right)$$ which shows the limit as how it is approached.
This would give good approximate values for finite values of $n$. For example $$S_{10}= \frac{155685007}{232792560}\approx 0.6687714$$ while $$\log(2)-\frac{1}{40}+\frac{1}{1600}\approx 0.6687722$$
