Lim -1p-2p-3p-np-np-1-AIEEE 2002-A- 1-p-1B- 1-1-pC- 1-p-1-p-1D- 1-P-2

The correct option is A lim_π→∞ 1^ρ+2^ρ+3^ρ+ ⋯ +n^ρ/n^ρ +1=lim_π→∞∑_r=1^n [ r^ρ/n^ρ +1 ] = lim_π→∞1/n∑_r=1^n ( r/n )^ρ = ∫_0^1 x^ρdx = [ x^ρ + 1/p + 1 ]^1_0 = ...

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Byju's Answer

Standard XII

Mathematics

Definite Integral as Limit of Sum

lim π→∞1p+2p+...

Question

$l i m_{π \to \infty} \frac{1^{ρ} + 2^{ρ} + 3^{ρ} + \dots + n^{ρ}}{n^{ρ + 1}} =$ [AIEEE 2002]

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Solution

The correct option is A

$l i m_{π \to \infty} \frac{1^{ρ} + 2^{ρ} + 3^{ρ} + \dots + n^{ρ}}{n^{ρ + 1}} = l i m_{π \to \infty} \sum_{r = 1}^{n} [\frac{r^{ρ}}{n^{ρ + 1}}]$
$= l i m_{π \to \infty} \frac{1}{n} \sum_{r = 1}^{n} {(\frac{r}{n})}^{ρ} = \int_{0}^{1} x^{ρ} d x = {[\frac{x^{ρ + 1}}{p + 1}]}_{0}^{1} = \frac{1}{p + 1}$

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lim n \to \infty \frac{1^{p} + 2^{p} + 3^{p} + . . . . . . + n^{p}}{n^{p + 1}}

is equal to (p

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limπ→∞ 1ρ+2ρ+3ρ+⋯+nρnρ+1= [AIEEE 2002]

The correct option is A limπ→∞ 1ρ+2ρ+3ρ+⋯+nρnρ+1=limπ→∞∑nr=1[rρnρ+1] =limπ→∞1n∑nr=1(rn)ρ=∫10 xρdx=[xρ+1p+1]10=1p+1

$l i m_{π \to \infty} \frac{1^{ρ} + 2^{ρ} + 3^{ρ} + \dots + n^{ρ}}{n^{ρ + 1}} =$ [AIEEE 2002]