JEE Advanced 2024 Paper 1 · Mathematics · Limits · easy

Let L=lim n→∞Σk=1^n(k^2/(n^3+k^3. The value of e^3L is ______.

Numerical answer

Let

L=\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\dfrac{k^2}{n^3+k^3}

. The value of

e^{3L}

is ______.

Show answer and solution

Answer: 2

Rewrite each term as

\dfrac{k^2}{n^3\left(1+\frac{k^3}{n^3}\right)}=\dfrac1n\cdot\dfrac{(k/n)^2}{1+(k/n)^3}

. The sum is a Riemann sum for

f(x)=\dfrac{x^2}{1+x^3}

[0,1]

, so

L=\displaystyle\int_0^1\dfrac{x^2}{1+x^3}\,dx

. Substituting

u=1+x^3

du=3x^2dx

, we get

L=\tfrac13\int_1^2\dfrac{du}{u}=\tfrac13\ln2

. Therefore

3L=\ln 2

and

e^{3L}=2

. Numerically, for

n=10^5

the sum is

0.231052

, matching

\tfrac13\ln2=0.231049

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