Question

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

CMR:1/2 < S <1

Answer 1

lớp mấy vay

Answer 2

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

Answer 3

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