S = (1/31+1/32+1/33+...+1/40) + (1/41 + 1/42 + ...+ 1/50) + (1/51 + 1/52+...+1/59+1/60)
Mà : (1/31+1/32+1/33+...+1/40) > 1/40 x 10 = 1/4 (gồm 10 số hạng) Tương tự : (1/41 + 1/42 + ...+ 1/50) > 1/5 ; (1/51 + 1/52+...+1/59+1/60) > 1/6 S > 1/4 + 1/5 + 1/6.
Trong khi đó (1/4 + 1/5 + 1/6) > 3/5 =>S > 3/5 (1) S = (1/31+1/32+1/33+...+1/40) + (1/41 + 1/42 + ...+ 1/50) + (1/51 + 1/52+...+1/59+1/60) Mà : (1/31+1/32+1/33+...+1/40) < 1/31 x 10 = 10/30 = 1/3 (gồm 10 số hạng) => S < 4/5 (2)
Từ (1) và (2) => 3/5 <S<4/5
\(S=\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+...+\frac{1}{50}\right)+\left(\frac{1}{51}+...+\frac{1}{60}\right)\)
Mà \(\left(\frac{1}{31}+...+\frac{1}{40}\right)>\frac{1}{40}\cdot10=\frac{1}{4}\)
Tương tự : \(\left(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}\right)>\frac{1}{5}\)
\(\left(\frac{1}{51}+...+\frac{1}{60}\right)>\frac{1}{6}\)
\(S>\frac{1}{4}+\frac{1}{5}+\frac{1}{6}>\frac{3}{5}\)(*1)
Mặt khác:\(\left(\frac{1}{31}+...+\frac{1}{40}\right)< \frac{1}{31}\cdot10=\frac{1}{3}\)
\(\Rightarrow S< \frac{4}{5}\)(*2)
Từ (*1)(*2)= 3/5<S<4/5
