Question

\( \int_0^1 \alpha^k x^k dx = \)

• A. \( \lim_{n \to \infty} \frac{\alpha^k(1 + 2^k + 3^k + \ldots + n^k)}{n^{k+1}} \)
• B. \( \lim_{n \to \infty} \frac{\alpha^k + \alpha^k + \ldots + \alpha^k}{n^{k+1}} \)
• C. \( \lim_{n \to \infty} \frac{1}{n} \sum \left( \frac{7}{n} \right)^k \)
• D. \( \lim_{n \to \infty} \frac{1}{n} \sum \left( \frac{27}{n} \right)^k \)

Hint:

Riemann sums approximate definite integrals using finite summation expressions — especially useful for limits!

Answer 1

The Correct Option is A

We consider Riemann sums for the integral:
\[ \int_0^1 \alpha^k x^k dx = \lim_{n \to \infty} \sum_{r=1}^n \alpha^k \left( \frac{r}{n} \right)^k \cdot \frac{1}{n} = \lim_{n \to \infty} \frac{\alpha^k}{n^{k+1}} \sum_{r=1}^n r^k \] \[ = \lim_{n \to \infty} \frac{\alpha^k(1 + 2^k + \ldots + n^k)}{n^{k+1}} \]