$$\begin{array}{ | c | c | } \hline n & \sum \limits_{k=1}^{n} \left(\frac{1}{x_k}\right)\left(\frac{1}{n}\right) \\ \hline 100 & 5.19 \\ \hline 200 & 5.88 \\ \hline 300 & 6.28 \\ \hline 400 & 6.47 \\ \hline 500 & 6.79 \\ \hline \hline \end{array}$$
The table above shows several Riemann sum approximations to $\int_0^1 \frac{1}{x}\, \mathrm{d}x$ using right-hand endpoints of $n$ subintervals of equal length of the interval $\left[0,1\right]$. Which of the following statements best describes the limit of the Riemann sums as $n$ approaches infinity?
(A) the limit of the Riemann sums is a finite number less than 10
(B) The limit of the Riemann sums is a finite number greater than 10
(C) The limit of the Riemann sums does not exist because $\left(\frac{1}{x_n}\right)\left(\frac{1}{n}\right)$ does not approach $0$.
(D) The limit of the Riemann sums does not exist because it is a sum of infinitely many positive number.
(E) The limit of the Riemann sums does not exist because $\int_0^1 \frac{1}{x}\, \mathrm{d}x$ does not exist. The correct answer
Now to my confusion
First, I understand the correct answer just from reading the problem. I know that if one takes a Riemann sum to infinity then it is the same as the Integral of the function. My problem is that --> I don't understand how to change the Riemann sum to an integral.<--
I don't want to just be able to answer the question, but to literally suck all the possible knowledge possible out of the question.
Actually your sum is in the form: $$\sum \limits_{k=1}^{n} \left(\frac{1}{x_k}\right)\left(\frac{1}{n}\right)=\sum \limits_{k=1}^{n} \left(\frac{1}{x_k}\right)\left(\frac{k+1}{n}-\frac{k}{n}\right)$$ Interpret the RHS as Integral i.e. $$\int f(x) dx$$ with $$dx=\left(\frac{k+1}{n}-\frac{k}{n}\right);\;\;f(x)=\left(\frac{1}{x_k}\right)$$ The bounds of the interval will be the largest and smallest value of $$\left(\frac{k}{n}\right)$$ i.e. 0 and 1 in your case.
It's worthwhile to recall the interpretation of Riemann Sum: $$\sum_{i=0}^{\infty}f{(t_i)}(x_{i+1}-x_i)$$ where $t_i$ is some intermediate value in the interval.