Abel's Test in Complex Analysis
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
and the series
converges when |z| < 1 and diverges when |z| > 1, and the coefficients {an} are positive real numbers decreasing monotonically toward the limit zero for n > m (for large enough n, in other words), then the power series for f(z) converges everywhere on the 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. Notice that Abel's test can also be applied to a power series with radius of convergence R ≠ 1 by a simple change of variables ζ = z/R.
Proof of Abel's test: Suppose that z is a point on the unit circle, z ≠ 1. Then
so that, for any two positive integers p > q > m, we can write
where Sp and Sq are partial sums:
But now, since |z| = 1 and the an are monotonically decreasing positive real numbers when n > m, we can also write
Now we can apply Cauchy's criterion to conclude that the power series for f(z) converges at the chosen point z ≠ 1, because sin(½θ) ≠ 0 is a fixed quantity, and aq+1 can be made smaller than any given ε > 0 by choosing a large enough q.
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