  In the classical literature on infinite series there are various tests to
determine if a given infinite series converges, diverges, or oscillates. But
unfortunately, for very many infinite series all the existing tests can fail to
provide definitive answers. In this article we propose a novel Bayesian theory
for assessment of convergence properties of any given infinite series.
Remarkably, this theory attempts to provide conclusive answers to the question
of convergence even where all the existing tests of convergence fail. We apply
our ideas to seven different examples, obtaining very encouraging results.
Importantly, we also apply our ideas to investigate the Riemann Hypothesis, and
obtain results that do not completely support the conjecture.
  We also extend our ideas to develop a Bayesian theory on oscillating series,
where we allow even infinite number of limit points. Analysis of Riemann
Hypothesis using Bayesian multiple limit points theory yielded almost identical
results as the Bayesian theory of convergence assessment.
