Question

[pluto]

so sánh 1/2^2+1/3^2+1/4^2+...+1/2013^2 và 2014/2013

Giúp mik với 

Answer 1

[ST]

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

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

............

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

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

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

Mà \(\frac{2014}{2013}>1\)

=> \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2013^2}< \frac{2014}{2013}\)