Question

Cho A=1/1^1+1/2^2+1/3^2+1/4^2+......+1/50^2. Chứng minh A<2.

Answer 1

\(\Rightarrow A<1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+.......+\frac{1}{49.50}\)

\(\Rightarrow A<1+\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.......+\frac{1}{49}-\frac{1}{50}\right)\)

\(\Rightarrow A<1+\left(1-\frac{1}{50}\right)\)

\(\Rightarrow A<1+\frac{49}{50}\)

\(\Rightarrow A<\frac{99}{50}\)

Vì \(\frac{99}{50}<2=\frac{100}{50}\Rightarrow A<2\)  ĐPCM

Answer 2

Ta có:

\(\frac{1}{2^2}<\frac{1}{1.2};\frac{1}{3^2}<\frac{1}{2.3};......;\frac{1}{50^2}<\frac{1}{49.50}\)

Do đó \(A=1+\frac{1}{2^2}+\frac{1}{3^2}+....+\frac{1}{50^2}<1+\frac{1}{1.2}+\frac{1}{2.3}+....+\frac{1}{49.50}\)

\(\Rightarrow A<1+\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{49}-\frac{1}{50}=2-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}=2-\frac{1}{50}<2\)

=>A<2(đpcm)