Question

Bài 4

CMR:1/2^2+1/3^2+...+1/2013^2<1

Answer 1

Ta có :

\(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2013^2}\)

\(< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2012.2013}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2012}-\frac{1}{2013}\)

\(=1-\frac{1}{2013}< 1\)( đpcm )

Answer 2

\(\frac{1}{2^2}< \frac{1}{1.2}=1-\frac{1}{2}\)

\(\frac{1}{3^2}< \frac{1}{2.3}=\frac{1}{2}-\frac{1}{3}\)

....

\(\frac{1}{2013^2}< \frac{1}{2012.2013}=\frac{1}{2012}-\frac{1}{2013}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2013^2}< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2012}-\frac{1}{2013}=1-\frac{1}{2013}< 1\)

Answer 3

Đặt \(A=\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2013^2}\)

Ta thấy \(\frac{1}{2^2}< \frac{1}{1.2};\frac{1}{3^2}< \frac{1}{2.3};...;\frac{1}{2013^2}\)\(< \frac{1}{2012.2013}\)

\(\Rightarrow A< B=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2012.2013}\)

MÀ \(B=1-\) \(\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...-\frac{1}{2012}+\frac{1}{2012}-\frac{1}{2013}\)

        \(=1-\frac{1}{2013}< 1\)

Mà A<B nên A<1(t/c)(đpcm)

Answer 4

ta thấy \(\frac{1}{2^2}\)<\(\frac{1}{1.2}\);\(\frac{1}{3^2}\)<\(\frac{1}{2.3}\);...;\(\frac{1}{2013^2}\)<\(\frac{1}{2012.2013}\)

\(\Rightarrow\)\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+...+\(\frac{1}{2013^2}\)<\(\frac{1}{1.2}\)+\(\frac{1}{2.3}\)+...+\(\frac{1}{2012.2013}\)=1-\(\frac{1}{2}\)+\(\frac{1}{2}\)-\(\frac{1}{3}\)+...+\(\frac{1}{2012}\)-\(\frac{1}{2013}\)=1-\(\frac{1}{2013}\)<1

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