Question

Cho s=1/3x3+1/4x4+1/5x5+...+1/20x20 Chứng tỏ s<1/2;s>1/4

 

Answer 1

\(S=\frac{1}{3.3}+\frac{1}{4.4}+\frac{1}{5.5}+...+\frac{1}{20.20}\)

Ta có:  \(\frac{1}{2}-\frac{1}{3}>\frac{1}{3.3}>\frac{1}{3}-\frac{1}{4}\)

           \(\frac{1}{3}-\frac{1}{4}>\frac{1}{4.4}>\frac{1}{4}-\frac{1}{5}\)

          \(\frac{1}{4}-\frac{1}{5}>\frac{1}{5.5}>\frac{1}{5}-\frac{1}{6}\)

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

           \(\frac{1}{19}-\frac{1}{20}>\frac{1}{20.20}>\frac{1}{20}-\frac{1}{21}\)

Cộng theo vế ta được:

\(\frac{1}{2}-\frac{1}{20}>S>\frac{1}{3}-\frac{1}{21}\)\(\Rightarrow\)\(\frac{1}{2}>S>\frac{1}{4}\)