lim _{n rightarrow infty} frac{left(1^2-1righ | vedclass

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(B) Let the numerator be $N = \sum_{r=1}^{n-1} (r^2-r)(n-r) = \sum_{r=1}^{n-1} (-r^3 + r^2(n+1) - nr)$.Using standard summation formulas:$N = -\left[\

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$\frac{1}{3}$

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(B) Let the numerator be $N = \sum_{r=1}^{n-1} (r^2-r)(n-r) = \sum_{r=1}^{n-1} (-r^3 + r^2(n+1) - nr)$.Using standard summation formulas:$N = -\left[\frac{(n-1)n}{2}\right]^2 + (n+1)\frac{(n-1)n(2n-1)}{6} - n\frac{(n-1)n}{2}$.Let the denominator be $D = \sum_{r=1}^n r^3 - \sum_{r=1}^n r^2 = \left[\frac{n(n+1)}{2}\right]^2 - \frac{n(n+1)(2n+1)}{6}$.As $n \rightarrow \infty$, the leading term of $N$ is $-\frac{n^4}{4} + \frac{2n^4}{6} = \frac{n^4}{12}$.The leading term of $D$ is $\frac{n^4}{4}$.Thus, $\lim _{n \rightarrow \infty} \frac{N}{D} = \frac{n^4/12}{n^4/4} = \frac{4}{12} = \frac{1}{3}$.

$\lim _{n \rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\ldots +\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\ldots +n^3\right)-\left(1^2+2^2+\ldots +n^2\right)}$ is equal to:

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