Câu 1:
\(B=\dfrac{5}{2.1}+\dfrac{4}{1.11}+\dfrac{3}{11.2}+\dfrac{1}{2.15}+\dfrac{13}{15.4}\)
\(\Rightarrow B=\dfrac{1}{7}\left(\dfrac{5}{2.7}+\dfrac{4}{7.11}+\dfrac{3}{11.14}+\dfrac{1}{14.15}+\dfrac{1}{15.28}\right)\)
\(\Rightarrow B=\dfrac{1}{7}\left(\dfrac{1}{2}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{14}+\dfrac{1}{14}-\dfrac{1}{15}+\dfrac{1}{15}-\dfrac{1}{28}\right)\)
\(\Rightarrow B=\dfrac{1}{7}\left(\dfrac{1}{2}-\dfrac{1}{28}\right)\)
\(\Rightarrow B=\dfrac{1}{7}.\dfrac{13}{28}\)
\(\Rightarrow B=\dfrac{13}{196}\)
Vậy \(B=\dfrac{13}{196}\)
Câu 2:
\(D=\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{20}}\)
\(\Rightarrow2D=1+\dfrac{1}{2}+...+\dfrac{1}{2^{19}}\)
\(\Rightarrow2D-D=\left(1+\dfrac{1}{2}+...+\dfrac{1}{2^{19}}\right)-\left(\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{20}}\right)\)
\(\Rightarrow D=1-\dfrac{1}{2^{20}}< 1\)
\(\Rightarrow D< 1\left(đpcm\right)\)
Vậy...