Answer to Question #140549 in Real Analysis for Nidhi

Question

Q.1 examine the convergence of series: 3×4/5^2 +'5×6/7^2 + 7×8/9^2 + ......

Answer 1

$\frac{3.4}{5^2}+\frac{5.6}{7^2}+\frac{7.8}{9^2}+\dots = \sum_{n=2}^\infin\frac{(2n-1).(2n)}{(2n+1)^2}= \sum_{n=2}^\infin U_n$

Then we have

$\lim_{n\to \infin} U_n= \lim_{n\to\infin}\frac{(2n-1).(2n)}{(2n+1)^2} = \lim_{n\to\infin}\frac{(2-\frac{1}{n}).2}{(2+\frac{1}{n})^2}= \frac{2.2}{4}=\frac{4}{4}=1 \ne0$

Since $\lim_{n\to \infin}U_n\ne0$ then by Cauchy's test for divergence the series is divergent(not convergent)