<s>[INST] Does the infinite product $\frac{2}{1}\times\frac{4}{3}\times\frac{6}{5}\times\dots$ converge? I was wondering whether the infinite product
$$\prod_{i=0}^{\infty}\frac{2n + 2}{2n + 1} = \frac{2}{1}\times\frac{4}{3}\times\frac{6}{5}\times\dots$$
converges. For all I know it is certainly not absolutely convergent, but is it at least conditionally so? Telescoping is not possible in this case. I know that
$$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots = \ln{2}$$
but if I take the logarithm off this infinite product, I'd get
$$ \ln{1} - \ln{\frac{1}{2}} + \ln{\frac{1}{3}} - \ln{\frac{1}{4}} + \dots$$
which is not conclusive.
 [/INST] Hint: If $a,b,c,\ldots$ are all positive 
$$(1+a)(1+b)(1+c)\cdots\ge 1+a+b+c+\cdots.$$
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