Question

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

Answer 1

Ta có: \(\frac{1}{1^2}=1\)

\(\frac{1}{2^2}=\frac{1}{4}\)

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

\(\frac{1}{4^2}< \frac{1}{3.4}=\frac{1}{3}-\frac{1}{4}\)

..........

\(\frac{1}{100^2}< \frac{1}{99.100}=\frac{1}{99}-\frac{1}{100}\)

\(\Rightarrow A< 1+\frac{1}{4}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)

\(\Rightarrow A< 1+\frac{1}{4}+\frac{1}{2}-\frac{1}{100}\)

\(\Rightarrow A< 1+\frac{3}{4}-\frac{1}{100}< 1+\frac{3}{4}\)