Question

Bài 1 :Cho A = 1/1^2 + 1/2^2 +1/3^2 +1/4^2+.....+1/50^2

Chứng minh A < 2

Bài 2 : S= 3 + 3/2 + 3/2^2 +.......+3/2^9

Answer 1

Bài 1:

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

\(< 1+\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{49.50}\)

\(=1+1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{49}-\dfrac{1}{50}\)

\(=1+1-\dfrac{1}{50}\)

\(=2-\dfrac{1}{50}\)

\(\Rightarrow A< 2-\dfrac{1}{50}< 2\)

\(\Rightarrow A< 2\left(đpcm\right)\)

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