To prove that $1+ \frac{1}{2} \frac{1}{3}+\frac{1}{2} \frac{3}{4} \frac{1}{5} + \frac{1}{2} \frac{3}{4} \frac{5}{6}\frac{1}{7}+.... $ converges.
If i attempt to find the nth term for this series i fail
I tired to make the denominator a factorial so that the series become like
$$1+ \frac{1}{3!} +\frac{3.3}{5!} +\frac{3.3.5.5}{7!} +\frac{3.3.5.5.7.7}{9!}+ .... $$ What would be the numerator of the nth term of this series
$$a_n = \frac{?}{(2n+1)!}, n=0,1,2,...$$
How does this series converges?
This is an exercise probelm 8 in methods of real analysis by Richard R goldberg, please help me with the proof.