lim-n-1-n-a-1-n-2-a-2-n-2-a-n-1-n-2-

Question Number 208093 by depressiveshrek last updated on 04/Jun/24

\lim_{n \to \infty} \frac{1}{n} ({(a + \frac{1}{n})}^{2} + {(a + \frac{2}{n})}^{2} + \dots + {(a + \frac{n - 1}{n})}^{2})

Answered by MM42 last updated on 04/Jun/24

= {l i m}_{n \to \infty} \frac{1}{n} \underset{i = 1}{\sum^{n - 1}} {(a + \frac{i}{n})}^{2}

{= \int_{0}^{1} {(a + x)}^{2} d x = \frac{1}{3} {(a + x)}^{3}]}_{0}^{1}

= \frac{1}{3} [{(a + 1)}^{3} - a^{3}]

= \frac{1}{3} (3 a^{2} + 3 a + 1) ✓

Commented by depressiveshrek last updated on 04/Jun/24

This was my way of doing it, sufficient

for a calculus 1 course

Commented by depressiveshrek last updated on 04/Jun/24