If (cn) n≥1 is a non-increasing sequence of positive real numbers tending to 0 in P. Bundschuh [Arch. Math. 29 (1977), 518–523] the question∞ was asked for which real numbers α the series (−1)[2nα]cn is convergent. For series of the form g(α) =∞ (−1)[2nα] it has be n e = n 1 shown in Bundschuh [ibid] that n n=1 it converges for numbers α with bounded continued fraction expansion and also for numbers like e, thereby solving a problem posed in H.D. Ruderman [Amer. Math. Monthly 83 (1977), 573]. Whether the series is convergent for α = π has∞ remained open. We could also deal with the more general series (−1)[2nα]cn, but then the technique would obscure the ideas of the proof. n=1 The author and S. Triˇckovi´c [J. Math. Anal. Appl. 342 (2006), 238–247] have proved that∞ (−1)[2nα] is convergent almost everywhere, that the function g(α) n n=1 is odd, has period 1 and represents a function in L2[0, 1]. Furthermore every open non empty interval contains two subsets P and N of the power of the continuum such that for α ∈ P we have g(α) = +∞ and for α ∈ N we have g(α) = −∞. If α = p is rational and p and q are coprime then the series is convergent if and q only if q is even. In this paper we determine all real numbers α for which the series g(α) is convergent.