let-give-S-n-1-i-lt-j-n-1-i-2-j-2-find-lim-n-gt-S-n-

Question Number 26749 by abdo imad last updated on 28/Dec/17

l e t g i v e S_{n} = \sum_{1 ⩽ i < j ⩽ n} \frac{1}{i^{2} j^{2}} f i n d {l i m}_{n - >\propto} S_{n} .

Commented by abdo imad last updated on 02/Jan/18

w e h a v e {(\sum_{k = 1}^{k = n} \frac{1}{k^{2}})}^{2} = \sum_{k = 1}^{n} \frac{1}{k^{4}} + 2 \sum_{1 ⩽ i < j ⩽ n} \frac{1}{i^{2} j^{2}}

S_{n} = \frac{1}{2} ({(\sum_{k = 1}^{n} \frac{1}{k^{2}})}^{2} - \sum_{k = 1}^{k = n} \frac{1}{k^{4}})

\Rightarrow {l i m}_{n - >\propto} S_{n} = \frac{1}{2} ({(ξ (2))}^{2} - ξ (4)) w e k n o w t h a t ξ (2) = \frac{π^{2}}{6}

a n d ξ (4) = \frac{π^{4}}{90}

\Rightarrow {l i m}_{n - >\propto} S_{n} = \frac{1}{2} (\frac{π^{4}}{36} - \frac{π^{4}}{90}) .