Mathematics

Problem

Evaluate the following : a.

(i) \lim_{n \to \infty} \left( \frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2} + \dots + \frac{n-1}{n^2} \right)

(ii) \lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n} \right)

(iii) \lim_{n \to \infty} \left( \frac{n}{n^2+1^2} + \frac{n}{n^2+2^2} + \dots + \frac{n}{2n^2} \right)

(iv) \lim_{n \to \infty} \frac{(1^p + 2^p + \dots + n^p)}{n^{p+1}}, p > 0