$\pi$ and $e$ as irrational linear combinations

Question

Let $S=\{m\cdot n^r\mid m,n\in\mathbb Z,r\in\mathbb Q\}$
Can $e$ or $\pi$ be written as a finite sum of elements of $S$?

Can $\pi=xe$, with $x$ algebraic?

Answer 1

As finite sums, no. If you could, you'll get that $\pi$ and $e$ are algebraic numbers, which is known to be false.

As infinite sums:

$$e = \sum_{n=0}^{\infty} \frac1{n!}$$

$$\pi = 4\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$$