ta có;1/11>1/20

         1/12>1/20

         1/13>1/20

        ................

       1/19>.1/20

cộng vế với vế của 1 và 2 ta đc

1/11+1/12+1/13+...+1/19>1/20+1/20+1/20+...+1/20

1/11+1/12+1/13+...+1/19+1/20>1/20+1/20+1/20+...+1/20+1/20[cộng cả 2 vế vs 1/20]

suy ra S>10/20

DO DÓ S>1/2

100% là đúng

s>1/2

vì sao ?

\(S=\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{20}>\frac{1}{20}+\frac{1}{20}+\frac{1}{20}+...+\frac{1}{20}=\frac{10}{20}=\frac{1}{2}\)

S = 1/11 + 1/12 + 1/13 + 1/14 + ... + 1/20

S > 1/20 + 1/20 + 1/20 + 1/20 + ... + 1/20

               10 phân số 1/20

S > 10 × 1/20

S > 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