Repair the scaled Taylor-series argument to prove irrationality of e^r for integers r>1

Determine whether the Taylor-series-based approach that assumes e^r = p/q, scales the series by q!, and attempts to bound the remainder term B = sum_{n=q+1}^\infty q! r^n / n! via a geometric-series estimate can be modified to yield 0 < B < 1 for integers r > 1, thereby establishing a contradiction and proving that e^r is irrational without appealing to orthogonal polynomials.

Background

The paper first adapts Fourier’s proof of the irrationality of e to attempt to prove the irrationality of e^r for a positive integer r by scaling the Taylor series by q! under the assumption e^r = p/q. This produces an integer part plus a remainder term B.

Bounding B with a geometric series works for r = 1 but fails for r > 1 because the bound becomes large for large q, and thus does not yield 0 < B < 1 needed for the contradiction. The author notes it is possible the argument could be repaired but does not know how, leaving the question open in this exposition.