Houston, we've had a product here

Calculus Level 2

If we take the product of all natural numbers \(a\) and their inverses \(a^{-1}\), we get 1, because we can use the one-to-one map \(\displaystyle a\rightarrow a^{-1}\) and therefore

\[\begin{align} \text{P } &=1\times\frac22 \times \frac33 \times \frac44 \ldots \\ &=\prod\limits_{a\in\mathbb{N}} aa^{-1} \\ &=\prod\limits_{a\in\mathbb{N}}1\\ &=1. \end{align}\]

If instead, we use the one-to-one map \(a\rightarrow \frac{1}{a-1}\), the product seems to be larger than 1, because every term in the product is greater than 1:

\[\begin{align} \text{P } &= 1\times 2 \times \left(\frac32 \times \frac43 \times \frac54\right) \ldots\\ \displaystyle &= 2\prod\limits_{a>2\in\mathbb{N}} \frac{a}{a-1} \\ &= \infty. \end{align}\]

Finally, if we pick \(a\rightarrow \frac{a}{a+1},\) the product seems to be zero.

Something is wrong. What's the deal?


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