Answer to Question #272701 in Real Analysis for Sourav

Question

Is (3x4)/5²+(5x6)/7²+(7x8)/9² +..... series convergence or not

Answer 1

$$\dfrac{3(4)}{5^2}+\dfrac{5(6)}{7^2}+\dfrac{7(8)}{9^2}+...=\displaystyle\sum_{n=1}^{\infin}\dfrac{(2n+1)(2n+2)}{(2n+3)^2}$$

$$a_n=\dfrac{(2n+1)(2n+2)}{(2n+3)^2}$$

$$\lim\limits_{n\to\infin}a_n=\lim\limits_{n\to\infin}\dfrac{(2n+1)(2n+2)}{(2n+3)^2}$$

$$=\lim\limits_{n\to\infin}\dfrac{(2n/n+1/n)(2n/n+2/n)}{(2n/n+3/n)^2}$$

$$=\lim\limits_{n\to\infin}\dfrac{(2+1/n)(2+2/n)}{(2+3/n)^2}$$

$$=\dfrac{(2+0)(2+0)}{(2+0)^2}=1\not=0$$

Then the series

$$\dfrac{3(4)}{5^2}+\dfrac{5(6)}{7^2}+\dfrac{7(8)}{9^2}+...=\displaystyle\sum_{n=1}^{\infin}\dfrac{(2n+1)(2n+2)}{(2n+3)^2}$$

diverges by the Test for Divergence.