đặt A=1/3²+1/4²+1/5²+……1/100²

B=1/2.3+1/3.4+...+1/99.100

=1/2-1/3...+1/99-1/100

=1/2-1/100<1/2 (1)

mà A=1/3²+1/4²+1/5²+……1/100²<B=1/2.3+1/3.4+...+1/99.100 (2)

kết hợp từ (1),(2)ta được A<B<1/2

=>A<1/2

\(\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+...+\frac{1}{2016^2}\)

\(=\frac{1}{2^2}.\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{1008^2}\right)< \frac{1}{2^2}.\left(1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{1007.1008}\right)\)

                                                                         \(< \frac{1}{4}.\left(1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{1007}-\frac{1}{1008}\right)\)

                                                                           \(< \frac{1}{4}.\left(2-\frac{1}{1008}\right)< \frac{1}{4}.2=\frac{1}{2}\)

=> đpcm

có: 1/3^2<1/2.3; 1/4^2<1/3.4:...: 1/100^2<1/99.100

Mà: 1/1.2+1/2.3+...+1/99.100=1-1/2+1/2-1/3+...+1/99-1/100

=1-1/100

=99/100

=> 1/3^2+1/4^2+...+1/100^2<99/100<1

=> đpcm

UNDERSTAND ???

đặt A= biểu thức trên

tao có 

A<1/2.3+1/3.4+...+1/99.100

A<1/2-1/3+1/3-1/4+...+1/99-1/100

A<1/2-1/100<1/2

SUY RA A<1/2(DPCM)