Chứng minh rằng: $\frac{1}{2^2}$ + $\frac{1}{3^2}$ + $\frac{1}{4^2}$ +…..+$\frac{1}{100^2}$ <1

Chứng minh rằng: $\frac{1}{2^2}$ + $\frac{1}{3^2}$ + $\frac{1}{4^2}$ +…..+$\frac{1}{100^2}$ <1

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  1. Ta thấy:

    $\frac{1}{2^{2} }$<$\frac{1}{1.2}$

    $\frac{1}{3^{2} }$< $\frac{1}{2.3}$

    $\frac{1}{4^{2} }$< $\frac{1}{3.4}$

                .               .               .

    $\frac{1}{100^{2} }$< $\frac{1}{99.100}$

    =>$\frac{1}{2^{2} }$+$\frac{1}{3^{2} }$+$\frac{1}{4^{2} }$+….+$\frac{1}{100^{2} }$<$\frac{1}{1.2}$+ $\frac{1}{2.3}$+$\frac{1}{3.4}$+…+$\frac{1}{99.100}$

    $\frac{1}{1.2}$+ $\frac{1}{2.3}$+$\frac{1}{3.4}$+…+$\frac{1}{99.100}$

    =1-$\frac{1}{2}$+$\frac{1}{2}$-$\frac{1}{3}$+$\frac{1}{3}$-$\frac{1}{4}$+…+$\frac{1}{99}$-$\frac{1}{100}$

    =1-$\frac{1}{100}$<1

    =>$\frac{1}{2^{2} }$+$\frac{1}{3^{2} }$+$\frac{1}{4^{2} }$+….+$\frac{1}{100^{2} }$<1

    Chúc bạn học tốt!!

       

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