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.
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[edit] Abel's test in real analysis
Given two sequences of real numbers, {an} and {bn}, if the sequences satisfy
-
converges
-
is monotonic and 
then the series
converges.
[edit] 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
![\begin{align}
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}}\,
\end{align}](../../../../math/e/0/d/e0df8f6d1b95ec9d960b330614ee6e8d.png)
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
![\begin{align}
\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}\,
\end{align}](../../../../math/5/3/f/53fd6d531bf5d028c3d28ad1fd71ac87.png)
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.
[edit] External links
[edit] Notes
- ^ (Moretti, 1964, p. 91)
[edit] References
- Gino Moretti, Functions of a Complex Variable, Prentice-Hall, Inc., 1964
