Infinite sum of products on my mid-term This problem was on my calculus mid-term : Determine the convergence or divergence of the   series  $$
\sum\limits_{n = 1}^\infty  {\prod\limits_{k = 1}^n  {\frac{{4k - 3}}{{4k - 1}}} } 
$$
I tried everything, even the Wallis product formula, but nothing worked out. I'm pretty sure it diverges but I can't find a way to prove that the general term doesn't tend to 0.
 The following question was (I'm only including this because it involves roughly the same series) :
Determine the convergence or divergence of the alternating series: $$
\sum\limits_{n = 1}^\infty  {( - 1)^n \prod\limits_{k = 1}^n  {\frac{{4k - 3}}{{4k - 1}}} } 
$$

Consider $$S_m=\sum\limits_{n = 1}^m  {\prod\limits_{k = 1}^n  {\frac{{4k - 3}}{{4k - 1}}} }=\sum\limits_{n = 1}^m u_n$$ with $$u_n=\prod\limits_{k = 1}^n  {\frac{{4k - 3}}{{4k - 1}}} $$ So $$\frac{u_{n+1}}{u_n}=\frac{(4n+1)}{(4n+3)}=1-\frac{2}{4 n+3}$$
I am sure that you can you take from here.
