Question

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

Answer 1

\(Cm:\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\)< 2

Ta có: \(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)

\(\Rightarrow\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< \frac{49}{50}< 1< 2\)

=> A < 2

tk nha mn

Answer 2

Ta có: \(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\) \(=1+\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)\) \(=1+\left(\frac{1}{2.2}+\frac{1}{3.3}+...+\frac{1}{50.50}\right)< 1+\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{50.51}\right)\)

\(\Rightarrow A< 1+\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{50}-\frac{1}{51}\right)\)

\(\Rightarrow A< 1+\left(\frac{1}{2}-\frac{1}{51}\right)=1+\frac{49}{102}< 1+1=2\) (Đpcm)

Answer 3

\(A=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}=1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\)

Ta có :\(\frac{1}{2^2}< \frac{1}{1.2}\)

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

            ................

           \(\frac{1}{50^2}< \frac{1}{49.50}\)

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

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

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

\(\Rightarrow A< 2\)(đpcm)