To test the convergence of series 1

To test the convergence of the following series:

1. $\displaystyle \frac{2}{3\cdot4}+\frac{2\cdot4}{3\cdot5\cdot6}+\frac{2\cdot4\cdot6}{3\cdot5\cdot7\cdot8}+...\infty $




2. $\displaystyle 1+ \frac{1^2\cdot2^2}{1\cdot3\cdot5}+\frac{1^2\cdot2^2\cdot3^2}{1\cdot3\cdot5\cdot7\cdot9}+ ...\infty $




3. $\displaystyle \frac{4}{18}+\frac{4\cdot12}{18\cdot27}+\frac{4\cdot12\cdot20}{18\cdot27\cdot36} ...\infty $

I cannot figure out the general $u_n$ term for these series(before I do any comparison/ratio test).

Any hints for these?

Answer

I cannot figure out the general $u_n$ term for these series(before I do any comparison/ratio test).

For the first series, one can start from the fact that, for every $n\geqslant1$, $$u_n=\frac{2\cdot4\cdots (2n)}{3\cdot5\cdots(2n+1)}\cdot\frac1{2n+2}=\frac{(2\cdot4\cdots (2n))^2}{2\cdot3\cdot4\cdot5\cdots(2n)\cdot(2n+1)}\cdot\frac1{2n+2},$$ that is, $$u_n=\frac{(2^n\,n!)^2}{(2n+1)!}\cdot\frac1{2n+2}=\frac{4^n\,(n!)^2}{(2n+2)!}.$$ Similar approaches yield the two other cases.