Infinite expression: $\frac{2}{1} \times \frac{2}{3} × \frac{4}{3} × \frac{4}{5} × \frac{6}{5}...$

Question

So there is this product $\frac{2}{1} \times \frac{2}{3} × \frac{4}{3} × \frac{4}{5} × \frac{6}{5}...$

I got its expression of a general term but converting it into a sum by a logarithm doesn't help either. Also could we independently say whether a limit exists or not as $4r^2>4r^2-1$ ?

Answer 1

Note you can write this as:

$$\prod_n \left(\frac {n+1}n\right)^{(-1)^{n-1}}.$$ Then the logarithm is $$\sum_n (-1)^{n-1}\log\left(1+\frac1n\right),$$ which converges since it is an alternating series that is decreasing in absolute value and converges to zero. https://en.wikipedia.org/wiki/Alternating_series_test

Answer 2

Note that $$ \ln\left(\frac k{k-1}\frac k{k+1}\right)=\ln \frac {k^2}{k^2-1}=\ln\left(1+\frac1{k^2-1}\right)<\frac1{k^2-1} $$ so that the series of logarithms converges