Question

CM : A=1/1.3 + 1/3.54 + ... + 1/(2n-1)(2n+1) < 1/2

Answer 1

\(A=\dfrac{1}{2}\left(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+...+\dfrac{2}{\left(2n-1\right)\left(2n+1\right)}\right)\)

\(=\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{2n-1}-\dfrac{1}{2n+1}\right)\)

\(=\dfrac{1}{2}\cdot\dfrac{2n+1-1}{2n+1}=\dfrac{n}{2n+1}\)

\(A-\dfrac{1}{2}=\dfrac{n}{2n+1}-\dfrac{1}{2}=\dfrac{2n-2n-1}{2\left(2n+1\right)}=\dfrac{-1}{2\left(2n+1\right)}< 0\)

=>A<1/2