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19 tháng 4 2019

Đặt \(A=1+\frac{1}{2}+...+\frac{1}{64}\)

Ta có: \(A=1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+...+\frac{1}{8}\right)+\left(\frac{1}{9}+...+\frac{1}{16}\right)+\left(\frac{1}{17}+...+\frac{1}{32}\right)\)\(+\left(\frac{1}{33}+...+\frac{1}{64}\right)\)

Ta thấy : \(\frac{1}{3}+\frac{1}{4}>\frac{1}{4}+\frac{1}{4}=\frac{1}{2}\)

                \(\frac{1}{5}+...+\frac{1}{8}>\frac{1}{8}+...+\frac{1}{8}=\frac{1}{8}.4=\frac{1}{2}\)

             \(\frac{1}{9}+\frac{1}{16}>\frac{1}{16}+...+\frac{1}{16}=\frac{1}{16}.8=\frac{1}{2}\)

             \(\frac{1}{17}+...+\frac{1}{32}>\frac{1}{32}+...+\frac{1}{32}=\frac{1}{32}.16=\frac{1}{2}\)

              \(\frac{1}{33}+...+\frac{1}{64}>\frac{1}{64}+...+\frac{1}{64}=\frac{1}{64}.32=\frac{1}{2}\)

\(\Rightarrow A>1+\frac{1}{2}.6=4\)

Vậy \(A>4\)

19 tháng 4 2019

\(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{63}+\frac{1}{64}>4.\)

Có :\(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{63}+\frac{1}{64}\)

\(=1+\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+...+\frac{1}{8}\right)+\left(\frac{1}{9}+...+\frac{1}{16}\right)+\left(\frac{1}{17}+....+\frac{1}{32}\right)+\left(\frac{1}{33}+...+\frac{1}{64}\right)\)

Ta có \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}=\frac{13}{12}\)

 và     \(\frac{1}{2}+\frac{1}{4}+\frac{1}{4}=1\)

\(\Rightarrow\frac{1}{2}+\frac{1}{3}+\frac{1}{4}>\frac{1}{2}+\frac{1}{4}+\frac{1}{4}\)

ta có  \(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}=\frac{533}{840}\)

và \(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{1}{2}\)

\(\Rightarrow\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}>\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\)

tương tự như trên ta tính được 

\(\frac{1}{9}+\frac{1}{10}+...+\frac{1}{16}>\frac{1}{16}\cdot8=\frac{1}{2}\)

\(\frac{1}{17}+\frac{1}{18}+...+\frac{1}{32}>\frac{1}{32}\cdot16=\frac{1}{2}\)

\(\frac{1}{33}+\frac{1}{34}+\frac{1}{35}+..+\frac{1}{64}>\frac{1}{64}\cdot32=\frac{1}{2}\)

\(\Rightarrow1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{63}+\frac{1}{64}>1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+1\)

\(\Rightarrow1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{63}+\frac{1}{64}>4\)

31 tháng 1 2016

S=1/4+1/9+1/16+1/25+1/36+1/49+1/64+1/81=1-1/81=1/81

 

31 tháng 1 2016

80/81 là đúng