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 Chúc bạn học tốt !

Tham khảo:

S=(1/31+1/32+1/33+...+1/40)+(1/41+1/42+1/43+...+1/50)+(1/51+1/52+1/53+...+1/60)"10 sống hạng mỗi ngoặc"

S<1/30 x 10+1/40 x 10+1/50 x 10

S<1/3+1/4+1/5=47/60<48/60=4/5

Học tốt~

Ta có :

\(S=\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{59}+\frac{1}{60}\) 

\(S=\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}\right)+\left(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60}\right)\)

\(\Rightarrow S>\left(\frac{1}{40}+\frac{1}{40}+...+\frac{1}{40}\right)+\left(\frac{1}{50}+\frac{1}{50}+...+\frac{1}{50}\right)+\left(\frac{1}{60}+\frac{1}{60}+...+\frac{1}{60}\right)\)

\(\Rightarrow S>\frac{1}{40}\cdot10+\frac{1}{50}\cdot10+\frac{1}{60}\cdot10\)

\(\Rightarrow S>\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\)

\(\Rightarrow S>\frac{37}{60}>\frac{36}{60}\) \(=\frac{3}{5}\)

\(\Rightarrow S>\frac{3}{5}\left(đpcm\right)\)

S = (1 / 31 + ... + 1 / 40) + (1 / 41 + ... + 1/ 50) + (1 / 51 + ... + 1 / 60) < 
10 / 31 + 10 / 41 + 10 / 51 < 10 / 30 + 10 / 40 + 10 / 50 = 1 / 3 + 1 / 4 + 1 / 5 = 
7 / 12 + 1 / 5 < 3 / 5 + 1 / 5 = 4 / 5 
Tương tự:
S > 10 / 40 + 10 / 50 + 10 / 60 = 1 / 4 + 1 / 5 + 1 / 6 = 5 / 12 + 1 / 5 > 2 / 5 + 1 / 5 = 3 / 5 
=> 3 / 5 < S < 4 / 5

Ta có S = \(\dfrac{1}{31}+\dfrac{1}{32}+\dfrac{1}{33}+...+\dfrac{1}{60}=\left(\dfrac{1}{31}+\dfrac{1}{32}+...+\dfrac{1}{40}\right)+\left(\dfrac{1}{41}+\dfrac{1}{42}+...+\dfrac{1}{50}\right)+\left(\dfrac{1}{51}+\dfrac{1}{52}+...+\dfrac{1}{60}\right)\)⇒ S < \(\dfrac{1}{30}\cdot10+\dfrac{1}{40}\cdot10+\dfrac{1}{50}\cdot10=\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}=\dfrac{47}{60}< \dfrac{48}{60}=\dfrac{4}{5}\)

Vậy S < \(\dfrac{4}{5}\)

\(S=\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{60}\)

\(\Rightarrow S=\left(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{45}\right)+\frac{1}{46}+\frac{1}{47}...+\frac{1}{60}\)

\(\Rightarrow S< \left(\frac{1}{30}+\frac{1}{30}+\frac{1}{30}+...+\frac{1}{30}\right)+\frac{1}{46}+\frac{1}{47}...+\frac{1}{60}\)(15 số hạng \(\frac{1}{30}\))

\(\Rightarrow S< \frac{15}{30}+\frac{1}{46}+\frac{1}{47}...+\frac{1}{60}< \frac{1}{2}< \frac{4}{5}\)

Vậy \(S< \frac{4}{5}\)

S < 1/40 x 30 = 3/4 < 4/5 

 =) S < 4/5

  Vậy S < 4/5

 học tốt nha

\(\Rightarrow\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{40}< \frac{1}{30}.10=\frac{1}{3}\left(1\right)\)

\(\Rightarrow\frac{1}{41}+\frac{1}{42}+\frac{1}{43}+...+\frac{1}{50}< \frac{1}{40}.10=\frac{1}{4}\left(2\right)\)

\(\Rightarrow\frac{1}{51}+\frac{1}{52}+\frac{1}{53}+...+\frac{1}{60}< \frac{1}{50}.10=\frac{1}{5}\left(3\right)\)

Từ (1) ,(2) và (3) suy ra

\(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{60}< \frac{1}{3}+\frac{1}{4}+\frac{1}{5}=\frac{47}{60}< \frac{48}{60}=\frac{4}{5}\)

\(\Rightarrow S< \frac{4}{5}\)

ta xét tổng của 1/31+...+1/40

tiếp tục 1/41+..+1/50

1/51+...+1/60

Trong 4 dãy số trên ta có 1/31> 1/32>1/33>...>1/41

=> Tổng trên < 10/31

cứ tiếp tục xét ta được S< 10/31+10/41+10/51<4/5

=> S < 4/5

Xét tương tự ta sẽ có S > 3/5