Browse Questions

# Evaluate $\lim_{n \rightarrow \infty} \bigg[ \large\frac{1^p + 2^p + 3^p....+ n^p}{n^{p+1}} \bigg ]$, where $p \gt -1$

(A) $p+1 \quad$ (B)$\large\frac{1}{p-1} \quad$ (C) $\large\frac{1}{p+1} \quad$ (D) $p-1$

$\lim_{n \rightarrow \infty} \bigg[ \large\frac{1^p + 2^p + 3^p....+ n^p}{n^{p+1}} \bigg ] = \large\frac{1}{n}$$\lim_{n \rightarrow \infty} \bigg[ \large\frac{1^p + 2^p + 3^p....+ n^p}{n^p} \bigg ] \quad = \large\frac{1}{n}$$ \lim_{n \rightarrow \infty} \bigg[ \bigg(\frac{1}{n}\bigg)^p + \bigg(\frac{2}{n}\bigg)^p + \bigg(\frac{3}{n}\bigg)^p + ... + \bigg(\frac{n}{n}\bigg)^p \bigg]$
Let $\large\frac{1}{n} $$= h \rightarrow nh = 1 and as n \rightarrow \infty, h \rightarrow 0 \large\frac{1}{n}$$ \lim_{n \rightarrow \infty} \bigg[ \bigg(\frac{1}{n}\bigg)^p + \bigg(\frac{2}{n}\bigg)^p + \bigg(\frac{3}{n}\bigg)^p + ... + \bigg(\frac{n}{n}\bigg)^p \bigg]$ $= \lim_{h \rightarrow 0} \bigg[ (1h)^p + (2h)^p + (3h)^p...+ (nh)^p \bigg]$
$\quad = \lim_{h \rightarrow 0} h \large \sum_{r=1}^{n}$$(rh)^p$$ = \Large \int_{\normalsize 0}^{\normalsize 1}$$x^p dx \quad = \bigg[ \large\frac{x^p + 1}{p+1} \bigg]_{0}^{1} \quad = \large\frac{1}{p+1} (when p \gt -1, this evaluates to \large\frac{1}{p+1}$$- 0$)