The value of lim n→∞ ∑ n^3 / (n^2 + k^2)(n^2 + 3k^2) [k=1, n] is:

Question

The value of \(\lim\limits _{n \rightarrow \infty} \sum_{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)}\) is:

(1) \(\frac{(2 \sqrt{3}+3) \pi}{24}\)

(2) \(\frac{13 \pi}{8(4 \sqrt{3}+3)}\)

(3) \(\frac{13(2 \sqrt{3}-3) \pi}{8}\)

(4) \(\frac{\pi}{8(2 \sqrt{3}+3)}\)

Answer 1

Correct option is (2) \(\frac{13 \pi}{8(4 \sqrt{3}+3)}\) 

\(\lim\limits _{n \rightarrow \infty} \sum_{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)} \)

\(\Rightarrow \lim\limits _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n \frac{1}{\left(1+\frac{k^2}{n^2}\right)\left(1+\frac{3 k^2}{n^2}\right)}\)

\(\Rightarrow \int_0^1 \frac{d x}{\left(1+x^2\right)\left(1+3 x^2\right)}=\frac{1}{2} \int_0^1 \frac{3\left(1+x^2\right)-\left(1+3 x^2\right)}{\left(1+x^2\right)\left(1+3 x^2\right)} d x\)

\(\Rightarrow \frac{1}{2}\left[\int_0^1 \frac{3 \mathrm{dx}}{1+3 \mathrm{x}^2}-\int_0^1 \frac{\mathrm{dx}}{1+\mathrm{x}^2}\right]=\frac{13 \pi}{8(4 \sqrt{3}+3)}\)