`Answer:`

 \(S=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{31}+\frac{1}{32}\)

a) Ta thấy:

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

\(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}>\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{1}{2}\)

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

\(\frac{1}{17}+\frac{1}{18}+...+\frac{1}{32}>16.\frac{1}{32}=\frac{1}{2}\)

\(\Rightarrow S>\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=\frac{5}{2}\)

b) Ta thấy:

\(\frac{1}{3}+\frac{1}{4}+\frac{1}{5}< 3.\frac{1}{3}\)

\(\frac{1}{6}+...+\frac{1}{11}< 6.\frac{1}{6}\)

\(\frac{1}{12}+...+\frac{1}{23}< 12.\frac{1}{12}\)

\(\frac{1}{24}+...+\frac{1}{32}< 9.\frac{1}{24}\)

\(\Rightarrow S< \frac{1}{2}+1+1+1+\frac{9}{24}=\frac{31}{8}< \frac{9}{2}\)

de lam ban oi!

ta có

1/2<1/1.2

1/3<1/2.3

...

1/32<1/31.32

=>1/2+1/3+...+1/32<1/1.2+1/2.3+...+1/31.32

=>1/2+1/3+...+1/32<1/1-1/2+1/2-1/3+...+1/31-1/32

=>1/2+1/3+...+1/32<1/1-1/32=31/32

vì 31/32<1

=>tổng đó <1

ta lại có 1+1=2 mà 2 <3

=>tổng đó <3

vậy:-------(bn tự lm nha)

k cho mik vs nha

đặt A = 1/31 + 1/32 + ... + 1/60

Tách A thành 3 nhóm ta được :

A = ( 1/31 + 1/32 + ... + 1/40 ) + ( 1/41 + 1/42 + ... + 1/50 ) + ( 1/51 + 1/52 + ... + 1/60 )

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

A < 1/3 + 1/4 + 1/5 = 47/60 < 48/60 = 4/5     ( đpcm )

Ta có: S=(1/31+1/32+...+1/40)+(1/41+1/42+...+1/50)+(1/51+1/52+...+1/60)

Mà: 1/31+1/32+...+1/40<1/31.10=10/30=1/3 (gồm 10 số hạng)

=> S<4/5

cảm ơn

bn tham khảo câu trả lời của mik ở bn hoàng thu trang nhabây h mik ghi lại dài dòng lắm

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

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

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

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

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