prove $$ 1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\dots+\dfrac{1}{n^2}

Question

prove $$ 1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\dots+\dfrac{1}{n^2}<2 $$

Uniteasy Admin · Accepted Answer

$$ 1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\dots+\dfrac{1}{n^2} $$
$$ <1+\dfrac{1}{2×1}+\dfrac{1}{3×2}+\dots+\dfrac{1}{n(n-1)} $$
$$ =1+\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dots+\dfrac{1}{n-1}-\dfrac{1}{n} $$
$$ =2-\dfrac{1}{n} $$
$$ <2 $$

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