Houston, we've had a product here

Calculus Level 2

If we take the product of all natural numbers aa and their inverses a1a^{-1}, we get 1, because we can use the one-to-one map aa1\displaystyle a\rightarrow a^{-1} and therefore

=1×22×33×44×=aNaa1=aN1=1.\begin{aligned} \text{P } &=1\times\frac22 \times \frac33 \times \frac44 \times \cdots \\ &=\prod\limits_{a\in\mathbb{N}} aa^{-1} \\ &=\prod\limits_{a\in\mathbb{N}}1\\ &=1. \end{aligned}

If instead, we use the one-to-one map a1a1a\rightarrow \frac{1}{a-1}, the product seems to be larger than 1, because every term in the product is greater than 1:

=1×2×(32×43×54)×=2a>2Naa1=.\begin{aligned} \text{P } &= 1\times 2 \times \left(\frac32 \times \frac43 \times \frac54\right) \times \cdots\\ \displaystyle &= 2\prod\limits_{a>2\in\mathbb{N}} \frac{a}{a-1} \\ &= \infty. \end{aligned}

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

Something is wrong. What's the deal?

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