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Answer by user17762 for Showing $\lim _{n\rightarrow \infty } \frac...

If $n$ is odd, then$$\sum_{i=1}^{(n+1)/2} i (n-i) = \frac{(n-1)(n+1)(n+3)}{12}$$Hence, you get that $$\frac{S_n}{n^2} = \frac{2}{n^2(n+1)} \times \frac{(n-1)(n+1)(n+3)}{12} =...

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Showing $\lim _{n\rightarrow \infty } \frac {S_{n}}{n^{2}} = \frac{1}{6}$

I am trying to show that if the arithmetic mean of the products of all distinct pairs of positive integers whose sum is $n$ is denoted by $S_{n}$ then $$\lim _{n\rightarrow \infty } \dfrac...

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