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 uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters.

Abel's test in real analysis[edit]

Suppose the following statements are true:

  1. \sum a_n is a convergent series,
  2. {bn} is a monotone sequence, and
  3. {bn} is bounded.

Then \sum a_nb_n is also convergent.

It is important to understand that this test is mainly pertinent and useful in the context of non absolutely convergent series \sum a_n. For absolutely convergent series, this theorem, albeit true, is almost evident.

Abel's test in complex analysis[edit]

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

\lim_{n\rightarrow\infty} a_n = 0\,

and the series

f(z) = \sum_{n=0}^\infty a_nz^n\,

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

z = e^{i\theta} \quad\Rightarrow\quad z^{\frac{1}{2}} - z^{-\frac{1}{2}} = 
2i\sin{\textstyle \frac{\theta}{2}} \ne 0

so that, for any two positive integers p > q > m, we can write

2i\sin{\textstyle \frac{\theta}{2}}\left(S_p - S_q\right) & = 
\sum_{n=q+1}^p a_n \left(z^{n+\frac{1}{2}} - z^{n-\frac{1}{2}}\right)\\
& = \left[\sum_{n=q+2}^p \left(a_{n-1} - a_n\right) z^{n-\frac{1}{2}}\right] -
a_{q+1}z^{q+\frac{1}{2}} + a_pz^{p+\frac{1}{2}}\,

where Sp and Sq are partial sums:

S_p = \sum_{n=0}^p a_nz^n.\,

But now, since |z| = 1 and the an are monotonically decreasing positive real numbers when n > m, we can also write

\left| 2i\sin{\textstyle \frac{\theta}{2}}\left(S_p - S_q\right)\right| & = 
\left| \sum_{n=q+1}^p a_n \left(z^{n+\frac{1}{2}} - z^{n-\frac{1}{2}}\right)\right| \\
& \le \left[\sum_{n=q+2}^p \left| \left(a_{n-1} - a_n\right) z^{n-\frac{1}{2}}\right|\right] +
\left| a_{q+1}z^{q+\frac{1}{2}}\right| + \left| a_pz^{p+\frac{1}{2}}\right| \\
& = \left[\sum_{n=q+2}^p \left(a_{n-1} - a_n\right)\right] +a_{q+1} + a_p \\
& = a_{q+1} - a_p + a_{q+1} + a_p = 2a_{q+1}.\,

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.

Abel's uniform convergence test[edit]

Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions or an improper integration of functions dependent on parameters. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of summation by parts.

The test is as follows. Let {gn} be a uniformly bounded sequence of real-valued continuous functions on a set E such that gn+1(x) ≤ gn(x) for all x ∈ E and positive integers n, and let {ƒn} be a sequence of real-valued functions such that the series Σƒn(x) converges uniformly on E. Then Σƒn(x)gn(x) converges uniformly on E.


  1. ^ (Moretti, 1964, p. 91)


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