Limit Of A Riemann Sum For An Improper Integral

Question

| $$n$$ | $$\displaystyle \sum_{k=1}^{n} \left(\frac{1}{x_k}\right)\left(\frac{1}{n}\right)$$ |
|-----|-------------------------------------------------------------|
| 100 | 5.19 |
| 200 | 5.88 |
| 300 | 6.28 |
| 400 | 6.57 |
| 500 | 6.79 |

The table shows the values of right Riemann sums with $$n$$ subintervals of equal width used to approximate $$\displaystyle \int_0^1 \frac{1}{x} \, dx$$. Which of the following statements best describes the limit of these Riemann sums as $$n$$ approaches infinity?

Accepted Answer

The limit of the Riemann sums does not exist because $$\displaystyle \int_0^1 \frac{1}{x} dx$$ does not exist.

Answer

The limit of the Riemann sums is a finite number less than 10.

Answer

The limit of the Riemann sums is a finite number greater than 10.

Answer

The limit of the Riemann sums does not exist because it is a sum of infinitely many positive numbers.

Limit Of A Riemann Sum For An Improper Integral

Question Leaderboard