`Answer:`

 \(S=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{31}+\frac{1}{32}\)

a) Ta thấy:

\(\frac{1}{3}+\frac{1}{4}>\frac{1}{4}+\frac{1}{4}=\frac{1}{2}\)

\(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}>\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{1}{2}\)

\(\frac{1}{9}+...+\frac{1}{16}>8.\frac{1}{16}=\frac{1}{2}\)

\(\frac{1}{17}+\frac{1}{18}+...+\frac{1}{32}>16.\frac{1}{32}=\frac{1}{2}\)

\(\Rightarrow S>\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=\frac{5}{2}\)

b) Ta thấy:

\(\frac{1}{3}+\frac{1}{4}+\frac{1}{5}< 3.\frac{1}{3}\)

\(\frac{1}{6}+...+\frac{1}{11}< 6.\frac{1}{6}\)

\(\frac{1}{12}+...+\frac{1}{23}< 12.\frac{1}{12}\)

\(\frac{1}{24}+...+\frac{1}{32}< 9.\frac{1}{24}\)

\(\Rightarrow S< \frac{1}{2}+1+1+1+\frac{9}{24}=\frac{31}{8}< \frac{9}{2}\)

S<1/2^2 + 1/2.3 + 1/3.4 +...+ 1/8.9

S<1/4 + 1/2 - 1/3 + 1/3 - 1/4+...+1/8 - 1/9

S<1/4 + 1/2 - 1/9

S<23/36<8/9 (1)

Mặt khác: S>1/2^2 + 1/3.4 + ...+ 1/9*10

S>1/4 + 1/3 - 1/4 + ... + 1/9 - 1/10

S>1/4 + 1/3 - 1/10

S>29/60>2/5 (2)

Từ (1),(2)

=> 2/5<S<8/9

Ta có S=1/2^2+1/3^2+1/4^2+...+1/9^2 
           <1/2²+1/2*3+1/3*4+....+1/8*9 
           =1/2²+1/2-1/3+1/3-1/4+....+1/8-1/9 
           =1/4+1/2-1/9=23/36<32/36=8/9 (♪) 
Ta lại có S=1/2^2+1/3^2+1/4^2+...+1/9^2 
                >1/2²+1/3*4+1/4*5+....+1/9*10 
                =1/2²+1/3-1/4+1/4-1/5+........+1/9-1/10 
                =1/2²+1/3-1/10 
                =19/20>8/20=2/5 ( ♫) 
                Từ (♪)( ♫) cho ta đpcm

Đpcm là j thế bạn

Điều phải chứng minh

ko biết

chứng minh rằng 1 nghĩa là sao bạn!?

chung minh lam sao cho may so tren =1

Ta có:
1/2^2 > 1/2.3
1/3^2 > 1/3.4
...
1/10^2 > 1/10.11
-> Cộng dọc theo vế ta có:
1/2^2+1/3^2+...+1/10^2 > 1/2.3+1/3.4+...+1/10.11
                                         = 1/2-1/3+1/3-1/4+...+1/10-1/11 

                                         = 1/2 - 1/11 = 9/22  (đpcm)