Question

CMR

\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{3}{4}\)

Answer 1

ta có : 1/2^2<1/2x3 

1/3^2<1/3x4

...........

1/100^2<1/99x100

suy ra :1/2^2+1/3^2 +........+1/100^2<1/2x3+1/3x4+1/4x5+..........+1/99x100

Gọi A=1/2x3+1/3x4+............+1/99x100

A=3-2/2x3+4-3/3x4+..........+100-99/99x100

A=3/2x3-2/2x3+4/3x4-3/3x4+........+100/99x100-99/99x100

A=1/2-1/100

A=49/100

1/2^2+1/3^2+......+1/100^2<49/100

Ta có:3/4=75/10049/100

Mà 75/100>49/100

1/2^2+1/3^2+........+1/100^2<3/4