# Houston, we've had a product here

**Calculus**Level 2

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 \(a\rightarrow \frac{a}{a+1}\) and the product seems to be zero.

Something is wrong. What's the deal?