calcul-la-somme-n-2-oo-x-n-2-1-2-

Question Number 181747 by SANOGO last updated on 29/Nov/22

c a l c u l l a s o m m e

\underset{n = 2}{\sum^{+ o o}} \frac{x}{{(n^{2} - 1)}^{2}}

Answered by Frix last updated on 01/Dec/22

\underset{n = 2}{\sum^{\infty}} \frac{x}{{(n^{2} - 1)}^{2}} = x \underset{n = 2}{\sum^{\infty}} \frac{1}{(n^{2} - 1)} =

= x \underset{n = 2}{\sum^{\infty}} (\frac{1}{4 {(n - 1)}^{2}} + \frac{1}{4 {(n + 1)}^{2}} - \frac{1}{2 (n^{2} - 1)}) =

= x (\frac{π^{2}}{24} + \frac{π^{2}}{24} - \frac{5}{16} - \lim_{k \to \infty} \frac{(k - 1) (3 k + 2)}{8 k (k + 1)}) =

= x (\frac{π^{2}}{12} - \frac{11}{16})

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