Transcendence of Series of Rational Functions and A Problem of Bundschuh

We investigate the transcendental nature of the sums

Σ f(n)A(n/B(n)

and

Σ A(n)/B(n)

where A(x), B(x) ∈ Q[x] with deg(A) < deg(B), f is an algebraic valued periodic function, and the sum is over integers n which are not zeros of B(x). We offer a new method of evaluating these sums using only elementary techniques. In some cases we relate these sums to a celebrated theorem of Nesterenko and a conjecture of Schneider and obtain concrete as well as conditional transcendence results. These results include progress on a problem of Bundschuh regarding the sum

Σ1/n^s-1

for integer values of s.