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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

## Abel's test in complex analysis

## External links

## Notes

## References

Given two sequences of real numbers, $\{a\_n\}$ and $\{b\_n\}$, if the sequences satisfy

- * $sum^\{infty\}\_\{n=1\}a\_n$ converges

- * $lbrace\; b\_n\; rbrace,$ is monotonic and $lim\_\{n\; rightarrow\; infty\}\; b\_n\; ne\; infty$

then the series

- $sum^\{infty\}\_\{n=1\}a\_n\; b\_n$

converges.

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 {a_{n}} 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 S_{p} and S_{q} are partial sums:

- $$

But now, since |z| = 1 and the a_{n} 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 a_{q+1} can be made smaller than any given ε > 0 by choosing a large enough q.

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

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Last updated on Monday July 21, 2008 at 19:55:56 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday July 21, 2008 at 19:55:56 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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