$\frac{1}{p}+\frac{1}{q}=1$ vs $\sum_{n=0}^\infty \frac{1}{p^n}=q$
In the study of $L^p$ spaces, I think this implies, e.g., that a particular generalization of Hölder's inequality holds: Let $f_n \in L^{p^n}$ for $n \in \mathbb N$. Then
$$\left\Vert \prod_{n=0}^\infty f_n\right\Vert_{1/q} \leq \prod_{n=0}^\infty \Vert f_n \Vert_{p^n}.$$
This can be proven by passing to the limit here. I'm sure that one can construct more similarly artificial applications of this result.
Edit: It turns out that what I stated here is actually a special case of this answer.