Abel's test

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In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Abel. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis.

Abel's test in real analysis

Given two sequences of real numbers, and , if the sequences satisfy

  • converges
  • is monotonic and

then the series


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.[1]

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.

External links


  1. (Moretti, 1964, p. 91)


  • Gino Moretti, Functions of a Complex Variable, Prentice-Hall, Inc., 1964

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