\(A=1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+......+\dfrac{1}{50^2}\)
Đặt :
\(I=\dfrac{1}{2^2}+\dfrac{1}{3^2}+.....+\dfrac{1}{50^2}\)
Ta có :
\(\dfrac{1}{2^2}< \dfrac{1}{1.2}\)
\(\dfrac{1}{3^2}< \dfrac{1}{2.3}\)
......
\(\dfrac{1}{50^2}< \dfrac{1}{49.50}\)
\(\Leftrightarrow\)\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+.....+\dfrac{1}{50^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+....+\dfrac{1}{49.50}\)
\(\Leftrightarrow I< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+....+\dfrac{1}{49}-\dfrac{1}{50}=1-\dfrac{1}{50}\)
\(\Leftrightarrow A=1+I< 1+1-\dfrac{1}{50}=2-\dfrac{1}{50}\)
\(\Leftrightarrow A< 2\left(đpcm\right)\)
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