  Power series are introduced that are simultaneously convergent for all real
and p-adic numbers. Our expansions are in some aspects similar to those of
exponential, trigonometric, and hyperbolic functions. Starting from these
series and using their factorial structure new and summable series with
rational sums are obtained. For arguments $x\in {\mathbb Q}$ adeles of series
are constructed. Possible applications at the Planck scale are also considered.
