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If-n-gt-2-and-n-N-prove-1-2-2-1-3-2-1-n-2-lt-n-1-1-2-1-1-n-1-n-




Question Number 93315 by Shakhzod last updated on 12/May/20
If  n>=2  and nāŠ‚N .  prove (1/2^2 )+(1/3^2 )+...(1/n^2 )<=n(1āˆ’(1/2)(1+(1/n)))^(1/n)
$${If}\:\:{n}>=\mathrm{2}\:\:{and}\:{n}\subset\mathbb{N}\:. \\ $$$${prove}\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }+…\frac{\mathrm{1}}{{n}^{\mathrm{2}} }<={n}\left(\mathrm{1}āˆ’\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)\right)^{\frac{\mathrm{1}}{{n}}} \\ $$

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