A while ago I threw the following at a numerical evaluator (in the present case I'm using wolfram alpha)

$\prod_{v=2}^{\infty} \sqrt[v(v-1)]{v} \approx 3.5174872559023696493997936\ldots$

Recently, for exploratory reasons only, I threw the following product at wolfram alpha

$\prod_{n=1}^{\infty} \sqrt[n]{1+\frac{1}{n}} \approx 3.5174872559023696493997936\ldots$

(I have cut the numbers listed above off where the value calculated by wolfram alpha begins to differ)

Are these products identical or is there some high precision fraud going on here?

`$\frac{1}{v-1}-\frac{1}{v}=\frac{1}{v(v-1)}$`

.) $\endgroup$