Đáp án:
Giải thích các bước giải:
$\dfrac{1}{3^{2}}+\dfrac{1}{4^{2}}+...+\dfrac{1}{100^{2}}$
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Ta có: $\dfrac{1}{3^{2}}<\dfrac{1}{2.3}$ $;$ $\dfrac{1}{4^{2}}<\dfrac{1}{3.4}$ $;...;$ $\dfrac{1}{100^{2}}<\dfrac{1}{99.100}$
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$⇒\dfrac{1}{3^{2}}+\dfrac{1}{4^{2}}+...+\dfrac{1}{100^{2}}<\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{99.100}$
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$⇒\dfrac{1}{3^{2}}+\dfrac{1}{4^{2}}+...+\dfrac{1}{100^{2}}<\dfrac{1}{2}-\dfrac{1}{100}$
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$⇒\dfrac{1}{3^{2}}+\dfrac{1}{4^{2}}+...+\dfrac{1}{100^{2}}<\dfrac{49}{100}<\dfrac{1}{2}$
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$⇒\dfrac{1}{3^{2}}+\dfrac{1}{4^{2}}+...+\dfrac{1}{100^{2}}<\dfrac{1}{2}$
