Question

Chứng minh rằng :

1 / 2² + 1 / 3² + 1/ 4² + 1/ 5^{2 + .... +} 1/ 100² < 1

Answer 1

ta thấy :

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

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

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

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

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

=\(\frac{99}{100}\)<1

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