Conditional Convergence of Infinite Products
William F. Trench · American Mathematical Monthly · 1999
In this article we revisit the classical subject of infinite products. For standard definitions and theorems on this subject see [1] or almost any textbook on complex analysis. We will restate parts of this material required to set the stage for our results, as follows. The infinite product P = HI(1 + a,,) of complex numbers is said to converge if there is an integer N such that 1 + an + 0 for n ? N and limnfl fnl_N(1 + a,n) is finite and nonzero. This occurs if and only if the series ,' Nlog(1 + a,,) converges. We say that P converges absolutely if H'(1 + la, I) converges. If P converges absolutely then P converges, but the converse is false. The following theorem [1, p. 223] settles the question of absolute convergence of infinite products.