Đáp án:
Giải thích các bước giải:
$A=1+\dfrac{1}{2^{2}}+\dfrac{1}{3^{2}}+...+\dfrac{1}{100^{2}}$
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Ta có: $\dfrac{1}{2^{2}}<\dfrac{1}{1.2}$ $;$ $\dfrac{1}{3^{2}}<\dfrac{1}{2.3}$ $;...;$ $\dfrac{1}{100^{2}}<\dfrac{1}{99.100}$
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$⇒A=1+\dfrac{1}{2^{2}}+\dfrac{1}{3^{2}}+...+\dfrac{1}{100^{2}}<1+\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{99.100}$
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$⇒A=1+\dfrac{1}{2^{2}}+\dfrac{1}{3^{2}}+...+\dfrac{1}{100^{2}}<1+1-\dfrac{1}{100}$
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$⇒A=1+\dfrac{1}{2^{2}}+\dfrac{1}{3^{2}}+...+\dfrac{1}{100^{2}}<2-\dfrac{1}{100}<2$
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$⇒A=1+\dfrac{1}{2^{2}}+\dfrac{1}{3^{2}}+...+\dfrac{1}{100^{2}}<2$ (đpcm)
