Prove-that-1-1-4-1-9-1-n-2-lt-2-for-n-N-

Question Number 159497 by naka3546 last updated on 17/Nov/21

P r o v e t h a t

1 + \frac{1}{4} + \frac{1}{9} + \dots + \frac{1}{n^{2}} < 2

f o r n \in N .

Commented by mr W last updated on 18/Nov/21

1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \dots + \frac{1}{n^{2}}

< 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \dots + \frac{1}{n^{2}} + \frac{1}{{(n + 1)}^{2}} + \dots = \frac{π^{2}}{6}

< \frac{10}{6} < \frac{12}{6} = 2

\Rightarrow 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \dots + \frac{1}{n^{2}} < 2

Commented by naka3546 last updated on 18/Nov/21

T h a n k y o u, s i r .

Commented by SANOGO last updated on 18/Nov/21

p r k w \frac{π^{2}}{6} e x p l i q u e m o i u n p e u

Commented by mr W last updated on 18/Nov/21

https://en.m.wikipedia.org/wiki/Basel_problem