Gọi \(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{63}\) là \(S\)
\(S=1+\dfrac{1}{2}+\left(\dfrac{1}{3}+\dfrac{1}{4}\right)+\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}\right)+\left(\dfrac{1}{9}+\dfrac{1}{10}+...+\dfrac{1}{16}\right)+\left(\dfrac{1}{17}+\dfrac{1}{18}+...+\dfrac{1}{32}\right)+\left(\dfrac{1}{33}+\dfrac{1}{34}+...+\dfrac{1}{63}+\dfrac{1}{64}\right)-\dfrac{1}{64}\\ =\left(1-\dfrac{1}{64}\right)+\dfrac{1}{2}+\left(\dfrac{1}{3}+\dfrac{1}{4}\right)+\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}\right)+\left(\dfrac{1}{9}+\dfrac{1}{10}+...+\dfrac{1}{16}\right)+\left(\dfrac{1}{17}+\dfrac{1}{18}+...+\dfrac{1}{32}\right)+\left(\dfrac{1}{33}+\dfrac{1}{34}+...+\dfrac{1}{63}+\dfrac{1}{64}\right)\)
Ta nhận thấy:
\(\dfrac{1}{3}\) lớn hơn \(\dfrac{1}{4}\)
\(\dfrac{1}{5},\dfrac{1}{6},\dfrac{1}{7}\) đều lớn hơn \(\dfrac{1}{8}\)
\(\dfrac{1}{9},\dfrac{1}{10},...,\dfrac{1}{15}\) đều lớn hơn \(\dfrac{1}{16}\)
\(\dfrac{1}{17},\dfrac{1}{18},...,\dfrac{1}{31}\) đều lớn hơn \(\dfrac{1}{32}\)
\(\dfrac{1}{33},\dfrac{1}{34},...,\dfrac{1}{63}\) đều lớn hơn \(\dfrac{1}{64}\)
\(\Rightarrow S>\left(1-\dfrac{1}{64}\right)+\dfrac{1}{2}+\left(\dfrac{1}{4}+\dfrac{1}{4}\right)+\left(\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}\right)+\left(\dfrac{1}{16}+\dfrac{1}{16}+...+\dfrac{1}{16}\right)+\left(\dfrac{1}{32}+\dfrac{1}{32}+...+\dfrac{1}{32}\right)+\left(\dfrac{1}{64}+\dfrac{1}{64}+...+\dfrac{1}{64}\right)\\ S>\left(1-\dfrac{1}{64}\right)+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}\\ S>\dfrac{63}{64}+\left(\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}\right)\\ S>\dfrac{63}{64}+3>3\)Mặt khác ta có:
\(S=1+\left(\dfrac{1}{2}+\dfrac{1}{3}\right)+\left(\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}\right)+\left(\dfrac{1}{8}+\dfrac{1}{9}+...+\dfrac{1}{15}\right)+\left(\dfrac{1}{16}+\dfrac{1}{17}+...+\dfrac{1}{31}\right)+\left(\dfrac{1}{32}+\dfrac{1}{33}+...+\dfrac{1}{63}\right)\)
\(\dfrac{1}{3}\) bé hơn \(\dfrac{1}{2}\)
\(\dfrac{1}{5},\dfrac{1}{6},\dfrac{1}{7}\) đều bé hơn \(\dfrac{1}{4}\)
\(\dfrac{1}{9},\dfrac{1}{10},...,\dfrac{1}{15}\) đều bé hơn \(\dfrac{1}{8}\)
\(\dfrac{1}{17},\dfrac{1}{18},...,\dfrac{1}{31}\) đều bé hơn \(\dfrac{1}{16}\)
\(\dfrac{1}{33},\dfrac{1}{34},...,\dfrac{1}{63}\) đều bé hơn \(\dfrac{1}{32}\)
\(\Rightarrow S< 1+\left(\dfrac{1}{2}+\dfrac{1}{2}\right)+\left(\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}\right)+\left(\dfrac{1}{8}+\dfrac{1}{8}+...+\dfrac{1}{8}\right)+\left(\dfrac{1}{16}+\dfrac{1}{16}+...+\dfrac{1}{16}\right)+\left(\dfrac{1}{32}+\dfrac{1}{32}+...+\dfrac{1}{32}\right)\\ S< 1+1+1+1+1+1\\ S< 6\)
