Is there a formula to calculate Σ_(i=1) ^n (1/i^2 ) interms of n..?


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Question Number 88000 by jdmath last updated on 07/Apr/20

$I s t h e r e a f o r m u l a t o c a l c u l a t e$ $\underset{i = 1}{\sum^{n}} \frac{1}{i^{2}}$ $i n t e r m s o f n . . ?$
Commented by abdomathmax last updated on 08/Apr/20

$i t h n k ? n o ? f o r m u l a e b u t w e c a n w r i t e$ $\sum_{i = 1}^{n} \frac{1}{i^{2}} = \sum_{i = 3 k} (. . .) + \sum_{i = 3 k + 1} (. . .) + \sum_{i = 3 k + 2} (. . .)$ $= \sum_{k = 1}^{[\frac{n}{3}]} \frac{1}{9 k^{2}} + \sum_{k = 0}^{[\frac{n - 1}{3}]} \frac{1}{{(3 k + 1)}^{2}} + \sum_{k = 0}^{[\frac{n - 2}{3}]} \frac{1}{{(3 k + 2)}^{2}}$ $= . . . .$
Commented by jdmath last updated on 16/Apr/20

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