Question

Chứng minh rằng : 1/2^2+1/3^2+1/4^2+....+1/100^2

Answer 1

Ta có:\(\frac{1}{2^2}<\frac{1}{1.2}\)

\(\frac{1}{3^2}<\frac{1}{2.3}\)

..........

\(\frac{1}{100^2}<\frac{1}{99.100}\)

=>\(\frac{1}{2^2}+\frac{1}{3^2}+.........+\frac{1}{100^2}<\frac{1}{1.2}+\frac{1}{2.3}+...........+\frac{1}{99.100}\)

Mà \(\frac{1}{1.2}+\frac{1}{2.3}+............+\frac{1}{99.100}\)

=\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}.............+\frac{1}{99}-\frac{1}{100}\)

=\(1-\frac{1}{100}<1\)

Vì \(\frac{1}{2^2}+\frac{1}{3^2}+...........+\frac{1}{100^2}<\frac{1}{1.2}+\frac{1}{2.3}+...........+\frac{1}{99.100}\)nên 

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

Answer 2

Thiếu đề rồi bạn Thủy ơi

Answer 3

mình quên xin lỗi 

Chứng minh rằng :1/2^2+1/3^2+1/4^2+...+1/100^2<1