lim_n →∞[ 1^3+2^3+3^3+........+ n^3n^4] = lim_n →∞1n[ ( 1n)^3+ #160; (2/n)^3 +........+ ( nn)^3] = lim_n →∞1n∑_n=1^r (r/n)^3 = ∫_0^1 x^3 d ...

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Byju's Answer

Standard XII

Mathematics

Integration of Piecewise Continuous Functions

limn→∞[13+23+...

Question

$lim n \to \infty [\frac{1^{3} + 2^{3} + 3^{3} + . . . . . . + n^{3}}{n^{4}}] =$

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Solution

$lim n \to \infty [\frac{1^{3} + 2^{3} + 3^{3} + . . . . . . . . + n^{3}}{n^{4}}]$

$= lim n \to \infty \frac{1}{n} [{(\frac{1}{n})}^{3} + (2 / n)^{3} + . . . . . . . . + {(\frac{n}{n})}^{3}]$

$= lim n \to \infty \frac{1}{n} r \sum n = 1 (r / n)^{3}$

$= \int_{0}^{1} x^{3} d x$ [using definition of definite integral ]
$= \frac{1}{4}$

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Similar questions

{lim}_{n \to \infty} \frac{1^{4} + 2^{4} + 3^{4} + . . . + n^{4}}{n^{5}} - {lim}_{n \to \infty} \frac{1^{3} + 2^{3} + 3^{3} + . . . + n^{3}}{n^{5}}

is:

Evaluate : ${lim}_{n \to \infty} \frac{1^{4} + 2^{4} + 3^{4} + . . . . + n^{4}}{n^{5}} - {lim}_{n \to \infty} \frac{1^{3} + 2^{3} . . . . + n^{3}}{n^{5}}$

{lim}_{n \to \infty} \frac{1^{3} + 2^{3} + . . . + n^{3}}{n^{4}} = \frac{1}{4} l .

Find

l

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

equals?

l i m_{n \to \infty} \frac{2^{3} - 1^{3}}{2^{3} + 1^{3}} . \frac{3^{3} - 1^{3}}{3^{3} + 1^{3}} . . . . \frac{n^{3} - 1}{n^{3} + 1}

equal

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limn→∞[13+23+33+......+n3n4]=

limn→∞[13+23+33+........+n3n4]=limn→∞1n[(1n)3+(2/n)3+........+(nn)3]=limn→∞1nr∑n=1(r/n)3=∫10x3dx [using definition of definite integral ]=14

$lim n \to \infty [\frac{1^{3} + 2^{3} + 3^{3} + . . . . . . + n^{3}}{n^{4}}] =$