Is my proof that πe is transcendental correct?

Question

I've been trying to prove that πe (the product of π and e) is transcendental. e can be defined as the infinite sum of reversed values of factorials (1/0!+1/1!+1/2!+1/3!+...). Hence, πe = π(1/0!+1/1!+1/2!+1/3!+...) = π/0!+π/1!+π/2!+π/3!+... Each number in this infinite series (π/0!, π/1! etc.) is transcendental (it's always π divided by an integer so it is transcendental because π is), so the sum of the infinite series (which equals πe) must be transcendental as well. Can anyone check my proof and say whether it is correct or not?

Answer 1

No, that argument is not correct. Being transcendental is not preserved under sums: e.g. $\pi+(-\pi)$ certainly isn't transcendental!

Infinite sums only ever make things worse, not better, so the fact that a finite sum of transcendentals isn't necessarily transcendental should convince you that your argument is incorrect; but just to drive the point home, here's an example of a sequence all of whose terms are transcendental (indeed, all of whose partial sums are transendental) whose sum is $1$: $$(1-\pi)+(\pi-{\pi\over 2})+({\pi\over 2}-{\pi\over 3})+({\pi\over 3}-{\pi\over 4})+...$$

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In general, negative properties like irrationality, transcendentality, etc. aren't preserved under sums - positive ones, like rationality, algebraicity, etc. tend to be.

Answer 2

Not to mention. As $\pi $ and $e $ both have infinite sum expansions then $1= e*e^{-1} = e^{-1}(\sum 1/n!)=\sum e^{-1}/n! $. The terms are transcendental.