$$\begin{array}{ | c | c | } \hline
n & \sum \limits_{k=1}^{n} \left(\frac{1}{x_k}\right)\left(\frac{1}{n}\right) \\ \hline
100 & 5.19 \\ \hline
200 & 5.88 \\ \hline
300 & 6.28 \\ \hline
400 & 6.47 \\ \hline
500 & 6.79 \\ \hline
\hline \end{array}$$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.
Best Answer
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.