A=1/31+1/32+...+1/149+1/150

1/31<1/30

1/32<1/30

...

1/40<1/30

1/41<1/40

1/42<1/40

...

1/50<1/40

...

1/140<1/130

1/141<1/140

...

1/150<1/140

=>A<10(1/30+1/40+...+1/140)

=>A<1/3+1/4+...+1/14=1,75<13/6

Ta có: S=1/31+1/32+1/33+...+1/60

=> 5S=5.(1/31+1/32+1/33+...+1/60)

 >5.(1/50+1/50+1/50+...+1/50) gồm (60-31):1+1=30 số 50

 =5.30/50=5.3/5=15/5=3

 Và 5S<5.(1/40+1/40+1/40+...+1/40) gồm 30 số 40

=5.30/40=5.3/4=15/4<16/4=4

 Vậy 3<5S<4

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

S có 30 số hạng.Nhóm thành ba nhóm, mỗi nhóm có 10 số hạng

\(S=\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+\frac{1}{42}+\frac{1}{43}+...+\frac{1}{50}\right)+\left(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60}\right)\)

\(S< \left(\frac{1}{30}+\frac{1}{30}+...+\frac{1}{30}\right)+\left(\frac{1}{40}+\frac{1}{40}+...+\frac{1}{40}\right)+\left(\frac{1}{50}+\frac{1}{50}+...+\frac{1}{50}\right)\)

\(S< \frac{10}{30}+\frac{10}{40}+\frac{10}{50}\)

\(S< \frac{47}{60}< \frac{50}{60}=\frac{5}{6}\)(1)

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

\(S>\frac{10}{40}+\frac{10}{50}+\frac{10}{60}\)

\(S>\frac{37}{60}>\frac{35}{60}\left(2\right)\)

Từ (1) và (2) => \(\frac{7}{12}< S< \frac{5}{6}\)

hay \(\frac{7}{12}< \frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{59}+\frac{1}{60}< \frac{5}{6}\)

Sửa cái phần đây nhá :  \(S>\frac{37}{60}>\frac{35}{60}=\frac{7}{12}\)