1but bhaiya r2x>=[r2x]>=r2x-1 ye hota hai na
r2x is always >= [r2x]
Q. Evaluate:
\lim_{n\rightarrow \infty}\frac{[1^{2}x]+[2^{2}x]+[3^{2}x]+ ...+[n^{2}x]}{n^{3}} , [.] \; is \; G.I.F
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4 Answers
106r^{2}x \leq [r^{2}x] < r^{2}x+1
=> \sum_{1}^{n}{r^{2}x} \leq \sum_{1}^{n}{[r^{2}x]} < \sum_{1}^{n}{r^{2}x+1}
=> \frac{nx(n+1)(2n+1)}{6} \leq \sum_{1}^{n}{[r^{2}x]} < \frac{nx(n+1)(2n+1)}{6}+n
=> \lim_{n\rightarrow \infty }\frac{nx(n+1)(2n+1)}{6n^{3}} \leq \lim_{n\rightarrow \infty }\frac{\sum_{1}^{n}{[r^{2}x]}}{n^{3}} < \lim_{n\rightarrow \infty }\frac{nx(n+1)(2n+1)}{6n^{3}}+\frac{1}{n^{2}}
=> \frac{x}{3} \leq \lim_{n\rightarrow \infty }\frac{\sum_{1}^{n}{[r^{2}x]}}{n^{3}} < \frac{x}{3}
So the required limit is \frac{x}{3}
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