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← JEE Main 2022 · 24 Jun · Shift 2
MathematicsSingle correct+4 / -1Question 7 of 90

JEE Main 2022 · 24 Jun · Shift 2, Mathematics question 7

lim⁡n→∞(n2(n2+1)(n+1)+n2(n2+4)(n+2)+n2(n2+9)(n+3)+⋯+n2(n2+n2)(n+n))\lim_{n\to\infty}\left(\frac{n^2}{\left(n^2+1\right)(n+1)}+\frac{n^2}{\left(n^2+4\right)(n+2)}+\frac{n^2}{\left(n^2+9\right)(n+3)}+\dots+\frac{n^2}{\left(n^2+n^2\right)(n+n)}\right)
is equal to

Choose one option
Answer and worked solution

Official answer: A

Riemann sum. The rr-th term is 1n⋅1(1+r2n2)(1+rn)\dfrac1n\cdot\dfrac1{\left(1+\frac{r^2}{n^2}\right)\left(1+\frac rn\right)}, so the limit is ∫01dx(1+x)(1+x2)\displaystyle\int_0^1\frac{dx}{(1+x)(1+x^2)}.

Partial fractions. 1(1+x)(1+x2)=1/21+x+12−12x1+x2\dfrac1{(1+x)(1+x^2)}=\dfrac{1/2}{1+x}+\dfrac{\frac12-\frac12x}{1+x^2}.

Integrate. 12log⁡e2+12⋅π4−14log⁡e2=π8+14log⁡e2\frac12\log_e2+\frac12\cdot\frac\pi4-\frac14\log_e2=\dfrac\pi8+\dfrac14\log_e2.

(A)

Topics:Limit of a sumPartial fractions
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