Transcendence in the affine case of Erdős Problem 270

For integers $a\geq1$ and $b\geq1-a$, the series $C_{a,b}=\sum_{n=1}^{\infty} n!/((a+1)n+b)!$ is transcendental. Equivalently, the series in Erdős Problem 270 is transcendental whenever $f(n)=an+b$ is a positive integer-valued affine function.

The manuscript claims $C_{a,b}$ is transcendental for every $a\ge1$ and $b\ge1-a$, settling the positive integer-valued affine subclass of Erdős Problem 270. Two pieces of context matter. Problem 270 as Erdős and Graham posed it, for every $f(n)\to\infty$, was already answered no by Crmarić and Kovač in 2025: for any $\alpha>0$ some such $f$ makes the series sum to $\alpha$. What survives is the non-decreasing case, and the affine family sits inside it. Separately, the checkable parts here were already known - the short irrationality proof for $C_{1,0}$ is Crmarić and Kovač's, posted by Kovač on the Erdős Problems forum in July 2026 and credited in the repository, and base-case transcendence follows a 2023 MathOverflow argument. The new content is the extension to the whole affine family, which is the part with neither formalization nor review.