Question

A=1/2^2 + 1/4^2 + 1/6^3 + ... +1/100^2 . CMR A<1/2

 

Answer 1

\(A = \dfrac{1}{2^2} + \dfrac{1}{4^2} +\dfrac{1}{6^2} +...... +\dfrac{1}{100^2} \)

\(A = \dfrac{1}{1^2.2^2} +\dfrac{1}{2^2.2^2} +\dfrac{1}{2^2.3^2} + .......+\dfrac{1}{2^2.2^{50}}\)

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

\(A < \dfrac{1}{2^2}.( \dfrac{1}{1.2}+\dfrac{1}{2.3}+......+\dfrac{1}{49.50}\) \()\)

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

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

\(< \dfrac{1}{2^2} . 2 = \dfrac{1}{2}\)