 A closely related convergence test, also known as Abel's test, can often be used to establish the convergence of a power series on the boundary of its circle of convergence.Specifically, Abel's test states that if a sequence of "positive real numbers" formula_4 is decreasing monotonically (or at least that for all "n" greater than some natural number "m", we have formula_5) with

then the power series

converges everywhere on the closed unit circle, except when "z" = 1.Abel's test cannot be applied when "z" = 1, so convergence at that single point must be investigated separately.