Question

a= 1/1²  + 1/2²   +1/3²   +1/4²    ...+1/50²

CHỨNG TỎ A < 2

Answer 1

Ta có :

1/2^ 2<1/1.2

1/3 ^2<1/2.3

1/4^ 2<1/3.4

.........................

1/50^ 2<1/49.50 

=>A<1/1 2+1/1.2+1/2.3+..+1/49.50

A<1+1-1/2+1/2=1/3+...+1/49-1/50 A<2-1/50<2(đpcm)

Answer 2

Ta có 1/2²<1/1.2

1/3²<1/2.3

1/4²<1/3.4

.........................

1/50²<1/49.50

=>A<1/1²+1/1.2+1/2.3+..+1/49.50

A<1+1-1/2+1/2=1/3+...+1/49-1/50

A<2-1/50<2(đpcm)

Answer 3

Ta có :

1/2^ 2<1/1.2

1/3 ^2<1/2.3

1/4^ 2<1/3.4

.........................

1/50^ 2<1/49.50 

=>A<1/1 2+1/1.2+1/2.3+..+1/49.50

A<1+1-1/2+1/2=1/3+...+1/49-1/50 A<2-1/50<2(đpcm)

Answer 4

Ta có :

1/2² < 1/1.2

1/3² < 1/2.3

1/4² < 1/3.4

.................

1/50² < 1/49.50

\(\Rightarrow\) A - 1/1² < 1/1.2 + 1/2.3 + 1/3.4 + ..........+ 1/49.50

      A - 1/1  < 1/1.2 + 1/2.3 + 1/3.4 + ...........+ 1/49.50

      A - 1 < 1/1.2 + 1/2.3 + 1/3.4 + .............+ 1/49.50

\(\Rightarrow\) A < 1 + 1/1.2 + 1/2.3 + 1/3.4 + ........... 1/49.50 ( hai vế cùng cộng với 1 )

      A < 1+ 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ........... + 1/49 - 1/50

      A < 1 + 1 - 1/50

      A < 2 - 1/50

Mà 2 - 1/50   <     2

\(\Rightarrow\) A < 2