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Limit \mathrm{\lim _{n \rightarrow \infty} \sum_{1 \leq i<j \leq n} \frac{(i+1)(j+1)}{n^4}} is

Option: 1

\frac{1}{4}


Option: 2

\frac{1}{6}


Option: 3

\frac{1}{8}


Option: 4

 none of these


Answers (1)

best_answer

\mathrm{\lim _{n \rightarrow \infty} \frac{\left(\sum_{i=1}^n(i+1)\right)^2=\sum_{i=1}^n(i+1)^2}{2 n^4}}
\mathrm{=\lim _{n \rightarrow \infty} \frac{\left(\frac{(n+2)(n+1)}{2}-1\right)^2-\left(\frac{(n+1)(n+2)(2 n+3)}{6}\right)}{2 n^4}=\frac{1}{8}}.

Posted by

Deependra Verma

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