Abel's test

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, $\{a\_n\}$ and $\{b\_n\}$, if the sequences satisfy

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

:* $lbrace\; b\_n\; brace,$ is monotonic and $lim\_\{n\; ightarrow\; infty\}\; b\_n\; e\; infty$

then the series

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

converges.

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

:$lim\_\{n\; ightarrowinfty\}\; a\_n\; =\; 0,$

and the series

:$f(z)\; =\; sum\_\{n=0\}^infty\; a\_nz^n,$

converges when |"z"| < 1 and diverges when |"z"| > 1, and the coefficients {"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". [(Moretti, 1964, p. 91)]

Proof of Abel's test: Suppose that "z" is a point on the unit circle, "z" ≠ 1. Then

:$z\; =\; e^\{i\; heta\}\; quadRightarrowquad\; z^\{frac\{1\}\{2\; -\; z^\{-frac\{1\}\{2\; =\; 2isin\{\; extstyle\; frac\{\; heta\}\{2\; e\; 0$

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

:

where "S"_"p" and "S"_"q" are partial sums:

:$S\_p\; =\; sum\_\{n=0\}^p\; a\_nz^n.,$

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(½"θ") ≠ 0 is a fixed quantity, and "a"_"q"+1 can be made smaller than any given "ε" > 0 by choosing a large enough "q".

External links

* [http://planetmath.org/encyclopedia/ProofOfAbelsTestForConvergence.html Proof (for real series) at PlanetMath.org]

Notes

References

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