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Source. Peter B. Borwein, On the irrationality of certain series, Mathematical Proceedings of the Cambridge Philosophical Society 112(1) (1992), 141--146, doi:10.1017/s030500410007081x. Theorem 2 is stated on p. 145; its proof runs pp. 145--146. Bibliographic details are on the source card.

Statement

Let $q$ be an integer with $|q|>1$ and let $c$ be a nonzero rational number with $c\ne-q^n$ for every $n$. Then

$$\sum_{n=1}^{\infty}\frac{(-1)^n}{q^n+c}$$

is irrational.

The introduction (p. 141) calls Theorem 2 new. As for Theorem 1, the claim that the number is not a Liouville number is argued only in the unnumbered closing paragraph on p. 146, as a sketch.

Proof sketch (pp. 145--146)

The paper gives only the points that differ from the proof of Theorem 1.

• The contour integral $F_n^*(q)$ inserts the sign $(-1)^h$ into the series of equation (1); residues express it through $\sum_{h\ge1}(-1)^h/(1-cq^h)$ plus a term from the pole at $t=0$, and it satisfies $|F_n^*(q)|\le 2\cdot2^n|c|^n/q^{3n^2/2}$ as printed (p. 145).
• Multiplying by $(n-2)!\prod_{k=1}^{n}(1-q^k)\prod_{k=1}^{n}(1-cq^k)\prod_{k=[n/3]}^{n}(1+q^k)$ gives a form $G_n(q)=\alpha_n(c,q)\sum_{h\ge1}(-1)^h/(1-cq^h)+\beta_n(c,q)$ with $\alpha_n,\beta_n$ having integer coefficients in $c$ and $q$; the factors $1+q^k$ come from the pole at zero (p. 145).
• The error estimate becomes $0<|G_n(q)|\le n!D^n/q^{n^2/18}$ as printed, for some constant $D=D_{q,c}$; nonvanishing is said to be essentially as in Lemma 5 (p. 146).

This sketch is written from a reading of the proof's structure; the paper's own proof is itself an outline, and the estimates were not re-derived here.

Read depth. Claims checked: the statement was read clause by clause on p. 145 of the printed article; the proof was read for structure only.

Dependencies

Theorem 1's proof, Lemmas 1--5 (pp. 142--144), which the proof adapts.

Bears on

• Problem 257: context only. With $q=2$ and $c=-1$ the series is the difference of the Problem 257 sums over the even and the odd positive integers; the theorem shows that difference is irrational but settles no set the problem asks about.