Ta thấy: \(\frac{1}{11}>\frac{1}{20}\)
\(\frac{1}{12}>\frac{1}{20}\)
\(\frac{1}{13}>\frac{1}{20}\)
................
\(\frac{1}{19}>\frac{1}{20}\)
=>\(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{19}+\frac{1}{20}>\frac{1}{20}+\frac{1}{20}+\frac{1}{20}+...+\frac{1}{20}+\frac{1}{20}\)
=>\(S>\frac{10}{20}\)
=>\(S>\frac{1}{2}\)(1)
Lại có:
\(\frac{1}{11}<\frac{1}{10}\)
\(\frac{1}{12}<\frac{1}{10}\)
\(\frac{1}{13}<\frac{1}{10}\)
................
\(\frac{1}{20}<\frac{1}{10}\)
=>\(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{20}<\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+...+\frac{1}{10}\)
=>\(S<\frac{10}{10}\)
=>S<1 (2)
Từ (1) và (2)
=>1/2<S<1
=>ĐPCM
Ta có:
1/20=1/20
1/19>1/20
1/18>1/20
.....
1/11>1/20
Suy ra 1/11+1/12+1/13+...+1/20>1/20*10=1/2. Vậy S>1/2.
Tương tự, ta có:
1/11=1/11
1/12<1/11
1/13<1/11
....
1/20<1/11
Suy ra 1/11+1/12+1/13+...+1/20<1/11*10=10/11<11/11=1. Vậy S < 1.
VẬY 1/2< S<1