-Ta có: \(\dfrac{1}{2}=\dfrac{10}{20}=\dfrac{1}{20}+\dfrac{1}{20}+...+\dfrac{1}{20}\) (có 10 số \(\dfrac{1}{20}\)).

Mà \(\dfrac{1}{20}< \dfrac{1}{19};\dfrac{1}{20}< \dfrac{1}{18};...;\dfrac{1}{20}< \dfrac{1}{11}\)

\(\Rightarrow\dfrac{1}{20}+\dfrac{1}{20}+...+\dfrac{1}{20}< \dfrac{1}{20}+\dfrac{1}{19}+\dfrac{1}{18}+...+\dfrac{1}{11}\)

\(\Rightarrow\dfrac{1}{2}< S\)

\(S=\frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}\)

\(>\frac{1}{20}+\frac{1}{20}+...+\frac{1}{20}\)(10 số hạng) 

\(=10.\frac{1}{20}=\frac{1}{2}\).

Vậy \(S>\frac{1}{2}\).

ta thấy: 1/11;1/12;1/13;...;1/19;1/20 đều >1/20

=>1/11+1/12+...1/19+1/20>1/20+1/20...+1/20

1/11+1/12+...1/19+1/20>10/20

1/11+1/12+...1/19+1/20>1/2 vậy S>1/2

vì 1/11>1/20

     1/12>1/20...

     1/13>1/20

nên 1/11+1/12+,,,,+1/20>1/20+1/20+,,,,+1/20=10/20=1/2(rút gọn

                                    10 số 1/20

vậy S>1/2

\(\frac{1}{11}\)> \(\frac{1}{20}\)

\(\frac{1}{12}\)> \(\frac{1}{20}\)

.

.

.

\(\frac{1}{19}\)>\(\frac{1}{20}\)

\(\frac{1}{20}\)= \(\frac{1}{20}\)

=> S  = 1/11+1/12+...+1/20>1/20+1/20+1/20+1/20+1/20+1/20+1/20+1/20+1/20+1/20=10*1/20=1/2 (đpcm)

Ta có:

\(\frac{1}{11}>\frac{1}{20}\)

\(\frac{1}{12}>\frac{1}{20}\)

.............

\(\frac{1}{20}=\frac{1}{20}\) 

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

\(\Leftrightarrow\frac{10.1}{20}=\frac{10}{20}=\frac{1}{2}\)

Vì \(\frac{1}{20}+\frac{1}{20}+...+\frac{1}{20}=\frac{1}{2}\). Mà \(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}>\frac{1}{20}\Rightarrow\frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}>\frac{1}{2}\)

Số lượng số của S là : 

\(\left(20-11\right):1+1=10\)( số ) 

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

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

\(\Rightarrow S>\frac{1}{20}.10\)

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

Vậy \(S>\frac{1}{2}\)

Ta có:

 1/11 + 1/12 + 1/13 + ....................... + 1/ 20 > 1/20 +1/20 +1/ 20 +1/20 +1/20 +1/20 +1/20 +1/ 20 +1/20 +1/20 = 1/2 

=> S > 1/2 

             Vậy S > 1/2

\(\frac{1}{11}>\frac{1}{20};\frac{1}{12}>\frac{1}{20};............;\frac{1}{20}=\frac{1}{20}\)    nên cái tổng ấy sẽ > 10 lần 1/20 =1/2

Ta thấy mỗi số hạng của tổng đều bé hơn 1/10

=>S<\(\frac{1}{10}.10=1\)

=>S<1

S = 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 + 1/17 + 1/18 + 1/19 + 1/20

S < 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10

S < 10 × 1/10

S < 1

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

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