Question

Prove that the series 1 + (1/2)·(1/3) + [(1·3)/(2· 4)]·(1/5) + [(1·3·5)/(2·4·6)]·(1/7) + ..... converges

Answer 1

We drop the first term i.e. 1. we consider the rest of the sequence Let us denote each term by $u_n.$ Then $u_1= \frac{1}{2}\cdot\frac{1}{3}$ . We note, $u_n= \frac{1}{2}\cdot\frac{3}{4}\cdots \frac{2n-1}{2n}\frac{1}{2n+1}.$ Hence $\frac{u_{n}}{u_{n+1}}=\frac{(2n+2)(2n+3)}{(2n+1)^2}.$

Hence, $n(\frac{u_{n}}{u_{n+1}}-1)= \frac{6n^2+5n}{4n^2+4n+1}=\frac{6+\frac{5}{n}}{4+\frac{4}{n}+\frac{1}{n^2}}.$ Hence $lim_{n\rightarrow \infty} \ n(\frac{u_n}{u_{n+1}}-1)=\frac{6}{4}=3/2>1.$

Hence by Rabbe's test the series converges.