Question

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

Steven Zheng Steven Zheng posted 1 month ago


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

Steven Zheng Steven Zheng posted 1 month ago

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