Question

Express the quantity
lim/n➡∞ 1/n ∑nk-1 [k/n]3

Answer 1

The goal is to represent the limit

\lim_{n\rightarrow \infty}\sum_{k=1}^n \left(1+\frac{2k}{n}\right)\cdot \frac{2}{n}

as an integral. Moy it help you

Answer 2

The quantity  $\lim_{n \to \infty} \frac{1}{n}∑^n_ k=1((\frac{k}{n} )^3)$ is expressed in integral form as follows:

Given:

The quantity  $\lim_{n \to \infty} \frac{1}{n}∑^n_ k=1((\frac{k}{n} )^3)$ is given.

To Find:

We have to find how the given quantity is expressed as an integral.

Solution:

The quantity, $\lim_{n \to \infty} \frac{1}{n}∑^n_ k=1((\frac{k}{n} )^3)$ is given.

Let $\frac{k}{n} = x$ and $\frac{1}{n} = dx.$

On applying the limits from infinity to n, we get the integral as:

$I = \int\limits^1_0 {x^{3} } \, dx$

$I = [\frac{x^{4} }{4}]^1_0\\\\i.e., I = \frac{1}{4} - 0\\\\\\ I = \frac{1}{4}.$

Hence, the quantity, $\lim_{n \to \infty} \frac{1}{n}∑^n_ k=1((\frac{k}{n} )^3)$ is expressed in integral form as $\frac{1}{4}.$

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