Câu hỏi của nguyenthingan

Question

chung minh rang 1/22+1/32+1/42+.......+1/20122<1

Hoàng Phúc · Accepted Answer

$\frac{1}{2^2}=\frac{1}{2.2}<\frac{1}{1.2};\frac{1}{3^2}=\frac{1}{3.3}<\frac{1}{2.3};...;\frac{1}{2012^2}=\frac{1}{2012.2012}<\frac{1}{2011.2012}$$=>\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2012^2}<\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+..+\frac{1}{2011.2012}$$=>\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2012^2}<\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+..+\frac{1}{2011}-\frac{1}{2012}=\frac{1}{1}-\frac{1}{2012}=\frac{2011}{2012}<1$=>đpcmCâu trả lời: $\frac{1}{2^2}=\frac{1}{2.2}<\frac{1}{1.2};\frac{1}{3^2}=\frac{1}{3.3}<\frac{1}{2.3};...;\frac{1}{2012^2}=\frac{1}{2012.2012}<\frac{1}{2011.2012}$$=>\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2012^2}<\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+..+\frac{1}{2011.2012}$$=>\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2012^2}<\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+..+\frac{1}{2011}-\frac{1}{2012}=\frac{1}{1}-\frac{1}{2012}=\frac{2011}{2012}<1$=>đpcm

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