Is the p-adic sum of an infinite polynomial with aperiodic coefficients necessarily irrational? I'm studying sequences with the form $bp^{a_1} + b^2p^{a_2}+b^3p^{a_3}\dotsc$, where $p$ is a $p$-adic base and $b$ is some rational number (edit:) and the exponents $(a_1, a_2, a_3,\dotsc)$ are strictly increasing.
It's easy to see that the p-adic sum of an infinite sequence with this form must be rational when the sequence of exponents  is periodic: $(a_1,a_2,\dotsc ,a_1+j, a_2+j, \dotsc ,a_1+2j, a_2+ 2j ,\dotsc)$, or even eventually periodic: that is, periodic after some finite number of terms. By analogy with $p$-adic expansions generally I feel that the converse must also be true - that  the sum of the sequence must be irrational when the exponents are aperiodic. Am I correct, and if so how can I prove it?
Every $p$-adic number can be written as a quotient of two $p$-adic integers, so in that sense, every $p$-adic number is “rational”. In fact, for $p &gt; 2$, $e^p \in \Bbb{Z}_p$, even though $e^p$ is not a rational number in $\Bbb{R}$.
Edit: @TorstenSchoeneberg in the comments tried to tell me that &quot;$e^p$&quot; (as defined by the usual convergent series $\sum_{n \geq 0} \frac{p^n}{n!}$) is not a number that exists in both $\Bbb{R}$ and $\Bbb{Q}_p$, because the metrics are different.
But the same logic he cites renders nonsensical his (and other people's) assertion that the OP is &quot;obviously&quot; looking for answers in &quot;$\Bbb{Q} \cap \Bbb{Q}_p$&quot;, since the type of infinite series OP is discussing, $$b p^{a_1} + b^2 p^{a_2} + b^3 p^{a_3} + ...$$ with $0 &lt; a_1 &lt; a_2 &lt; a_3 &lt; ...,$ does not even converge in $\Bbb{Q}$. Commenters who think the OP wants an answer in &quot;$\Bbb{Q} \cap \Bbb{Q}_p$&quot; (whatever that is) are wrong, and have failed to appreciate this subtle but essential distinction.
Every $p$-adic number can be written as a quotient of two $p$-adic integers, so in that sense, every $p$-adic number is “rational”. In fact, for $p &gt; 2$, $e^p \in \Bbb{Z}_p$, even though $e^p$ is not a rational number in $\Bbb{R}$.
Edit: @TorstenSchoeneberg in the comments tried to tell me that &quot;$e^p$&quot; (as defined by the usual convergent series $\sum_{n \geq 0} \frac{p^n}{n!}$) is not a number that exists in both $\Bbb{R}$ and $\Bbb{Q}_p$, because the metrics are different.
But the same logic he cites renders nonsensical his (and other people's) assertion that the OP is &quot;obviously&quot; looking for answers in &quot;$\Bbb{Q} \cap \Bbb{Q}_p$&quot;, since the type of infinite series OP is discussing, $$b p^{a_1} + b^2 p^{a_2} + b^3 p^{a_3} + ...$$ with $0 &lt; a_1 &lt; a_2 &lt; a_3 &lt; ...,$ does not even converge in $\Bbb{Q}$. Commenters who think the OP wants an answer in &quot;$\Bbb{Q} \cap \Bbb{Q}_p$&quot; (whatever that is) are wrong, and have failed to appreciate this subtle but essential distinction.