It just occurred to me that conjugate exponents, i.e. $p,q\in(1,+\infty)$ such that $$\frac{1}{p}+\frac{1}{q} =1$$ also satisfy the relations:

- $\sum_{n=0}^\infty \frac{1}{p^n}=q;$
- $\sum_{n=0}^\infty \frac{1}{q^n}=p.$

I never saw this fact used in the study of $L^p$ spaces... does anyone know any application of these relations in that context?