Diophantine Approximations and the Convergence of Certain Series

Consider two series Sigma(infinity)(n= 1) sin(n) pi theta n/n(alpha), Sigma(infinity)(n= 1) cos(n) pi theta n/n(alpha). We show that number-theoretical properties of. have a strong effect on the convergence when 0 < alpha <= 1. The complete investigation for theta is an element of Q is given. For irrational. we prove the result which depends on how well. can be approximated with rational numbers, i.e. on its irrationality measure. We obtain that if alpha > 1/2, then both series converge absolutely for almost all real.. Finally, we construct such an everywhere dense set of. that both series diverge when alpha <= 1.