What exactly is this limit question asking me to do?

Question

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!

Answer 1

$$\lim _{ n\to \infty } \left[ \frac { 1 }{ n+1 } +\frac { 1 }{ n+2 } +...+\frac { 1 }{ 2n } \right] =\lim _{ n\to \infty } \frac { 1 }{ n } \left[ \frac { 1 }{ 1+\frac { 1 }{ n } } +\frac { 1 }{ 1+\frac { 2 }{ n } } +...+\frac { 1 }{ 1+\frac { n }{ n } } \right] =\\ =\lim _{ n\to \infty } \frac { 1 }{ n } \sum _{ k=1 }^{ n }{ \frac { 1 }{ 1+\frac { k }{ n } } } =\int _{ 0 }^{ 1 }{ \frac { 1 }{ 1+x } dx } =\color{red}{\ln 2} $$

Answer 2

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 $x_k=\frac kn$.

Answer 3

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 $$\log(2)-\frac{1}{40}+\frac{1}{1600}\approx 0.6687722$$