S=1/2+1/3+1/4+1/5+1/6+1/7+1/8=

                                                     =481/280=1,717(857142)

                                                   => S<2

Ta có:\(\frac{1}{11}>\frac{1}{20};\frac{1}{12}>\frac{1}{20};\frac{1}{13}>\frac{1}{20};....;\frac{1}{19}>\frac{1}{20}\)

\(\Rightarrow\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{20}>\frac{1}{20}+\frac{1}{20}+\frac{1}{20}+...+\frac{1}{20}\)(Có 10 phân số \(\frac{1}{20}\))

\(\Rightarrow\frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}>\frac{10}{20}\)\(\Leftrightarrow S>\frac{10}{20}\)

Mà \(\frac{10}{20}=\frac{1}{2}\)nên

\(\Rightarrow S>\frac{1}{2}\)

A=(1/2+1/12+1/13+1/14+1/15)+(1/16+1/17+1/18+1/19+1/20)

Thay các phân số trong mỗi nhóm bởi phân số nhỏ nhất, ta có:

A> 1/15.5+1/20.5=1/3+1/4= 7/12>1/2

Suy ra A>1/2

Vậy A> 1/2

Vì 1/2<2/3;3/4<4/5;5/6<6/7;....;9999/10000<10000/10001

-->S<2/3.4/5.5/6.....10000/10001

Gọi 2/3.4/5.5/6.....10000/10001 là D và D>S

Có D=2/3.4/5.5/6.....10000/10001

-->S.D=(1/2 . 3/4 . 5/6.....9999/10000).(2/3.4/5.5/6.....10000/10001)

S.D=1/2.2/3.3/4.4/5.5/6.6.7.....9999/10000.10000/10001

S.D=1/10001

Vì S<D nên S.S<S.D hay S.S<1/10001

-->S<1/10001

mà 1/10001<1/100=0.01

-->S<1/100=0.01

-->S<0.01

Vậy

Có \(\frac{1}{2}< \frac{2}{3};\frac{2}{3}< \frac{3}{4};.....;\frac{9999}{10000}< \frac{10000}{10001}\)

\(\Rightarrow S^2< \frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot\cdot\cdot\cdot\cdot\frac{10000}{10001}\)

\(\Rightarrow S^2< \frac{1}{10001}< \frac{1}{10000}=\left(\frac{1}{100}\right)^2\)

\(\Rightarrow S< \frac{1}{1000}< 0,01\)

Vậy S<0,01

\(\frac{6}{8}< 1\)vì \(\frac{6}{8}< \frac{8}{8}\)

\(\frac{1}{4}< 1\)vì \(\frac{1}{4}< \frac{4}{4}\)

\(\frac{5}{2}>1\)vì \(\frac{5}{2}>\frac{2}{2}\)

\(\frac{7}{3}>1\)vì \(\frac{7}{3}>\frac{3}{3}\)

\(\frac{19}{19}=1\)vì \(\frac{19}{19}=\frac{19}{19}\)

Bài 1 :

S = \(\frac{6}{2.5}+\frac{6}{5.8}+...+\frac{6}{29.32}\)

   = 2 . \(\left(\frac{3}{2.5}+\frac{3}{5.8}+...+\frac{3}{29.32}\right)\)

   = 2 . \(\left(\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+...+\frac{1}{29}-\frac{1}{32}\right)\)

   = 2 . \(\left(\frac{1}{2}-\frac{1}{32}\right)\)= ....