Is-there-a-formula-to-calculate-i-1-n-1-i-2-interms-of-n-

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} (\dots) + \sum_{i = 3 k + 1} (\dots) + \sum_{i = 3 k + 2} (\dots)

= \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}}

= \dots .

Commented by jdmath last updated on 16/Apr/20

t h a n k y o u \dots r l y

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