A Constructive Study of Landau's Summability Theorem

A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined within Bishop-style constructive mathematics. It is shown that the original theorem is nonconstructive, and that a natural weakening of the theorem is constructively equivalent to Ishihara's principle BD-N. The paper ends with a number of results that, while not as strong as Landau's theorem, nevertheless contain positive computational information related to its conclusion.