S có 30 số hạng . Nhóm thành 3 nhóm , mỗi nhóm 10 số hạng.

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

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

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

\(S< \frac{37}{60}< \frac{48}{60}=\frac{4}{5}(1)\)

Lại có : \(S>\left[\frac{1}{40}+\frac{1}{40}+...+\frac{1}{40}\right]+\left[\frac{1}{50}+\frac{1}{50}+...+\frac{1}{50}\right]+\left[\frac{1}{60}+\frac{1}{60}+...+\frac{1}{60}\right]\)

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

\(S>\frac{37}{60}>\frac{36}{60}=\frac{3}{5}(2)\)

Từ 1 và 2 suy ra \(\frac{3}{5}< S< \frac{4}{5}\)

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Chúc hok tốt!!!!!!!!!!!!!!!

ta có B= 1/31+1/32+1/33+...+1/60

=> B=(1/30+1/30+...+1/30) + (1/40+1/40+1/40+...+1/40)

          10 số hạng                        10 số hạng

=> B< 10/30+10/40+10/50

=> = 1/3+1/4+1/5

=> = 47/60

=> B< 47/60 < 48/60= 4/5

Vế 2 tự làm nha bà

cần tau giúp ko con tê

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

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

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

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

\(S< \frac{37}{60}< \frac{48}{60}=\frac{4}{5}\)

\(S=\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+\frac{1}{34}+...+\frac{1}{60}\)\(CMR:S< \frac{4}{5}\)

Số số hạng của S là: (60 - 31 ) + 1 = 30 ( số ), chia thành 6 nhóm, mỗi nhóm 5 số hạng.

Ta có: 

\(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+\frac{1}{34}+\frac{1}{35}< \frac{1}{31}+\frac{1}{31}+\frac{1}{31}+\frac{1}{31}+\frac{1}{31}\)

\(\frac{1}{36}+\frac{1}{37}+\frac{1}{38}+\frac{1}{39}+\frac{1}{40}< \frac{1}{36}+\frac{1}{36}+\frac{1}{36}+\frac{1}{36}+\frac{1}{36}\)

\(\frac{1}{41}+\frac{1}{42}+\frac{1}{43}+\frac{1}{44}+\frac{1}{45}< \frac{1}{41}+\frac{1}{41}+\frac{1}{41}+\frac{1}{41}+\frac{1}{41}\)

\(\frac{1}{46}+\frac{1}{47}+\frac{1}{48}+\frac{1}{49}+\frac{1}{50}< \frac{1}{46}+\frac{1}{46}+\frac{1}{46}+\frac{1}{46}+\frac{1}{46}\)

\(\frac{1}{51}+\frac{1}{52}+\frac{1}{53}+\frac{1}{54}+\frac{1}{55}< \frac{1}{51}+\frac{1}{51}+\frac{1}{51}+\frac{1}{51}+\frac{1}{51}\)

\(\frac{1}{56}+\frac{1}{57}+\frac{1}{58}+\frac{1}{59}+\frac{1}{60}< \frac{1}{56}+\frac{1}{56}+\frac{1}{56}+\frac{1}{56}+\frac{1}{56}\)

\(=>S=\frac{5}{31}+\frac{5}{36}+\frac{5}{41}+\frac{5}{46}+\frac{5}{51}+\frac{5}{56}\)

\(=>S< 0,78...\)\(=>S< \frac{7}{10}\)( mình ước lượng thôi nha )

Vậy \(S< \frac{4}{5}\)vì \(\frac{4}{5}=\frac{8}{10}< \frac{7}{10}\)

~UMK..., mình ko chắc đúng ko nữa~

\(\frac{3}{5}< S< \frac{4}{5}\)mới đúng 

chớ sao lại \(3< 5S< 4\)