- 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 withpower series incomplex analysis .Abel's test in real analysis
Given two
sequence s ofreal number s, 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
:egin{align}2isin{ extstyle frac{ heta}{2left(S_p - S_q ight) & = sum_{n=q+1}^p a_n left(z^{n+frac{1}{2 - z^{n-frac{1}{2 ight)\& = left [sum_{n=q+2}^p left(a_{n-1} - a_n ight) z^{n-frac{1}{2 ight] -a_{q+1}z^{q+frac{1}{2 + a_pz^{p+frac{1}{2,end{align}
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
:egin{align}left| 2isin{ extstyle frac{ heta}{2left(S_p - S_q ight) ight| & = left| sum_{n=q+1}^p a_n left(z^{n+frac{1}{2 - z^{n-frac{1}{2 ight) ight| \& le left [sum_{n=q+2}^p left| left(a_{n-1} - a_n ight) z^{n-frac{1}{2 ight| ight] +left| a_{q+1}z^{q+frac{1}{2 ight| + left| a_pz^{p+frac{1}{2 ight| \& = left [sum_{n=q+2}^p left(a_{n-1} - a_n ight) ight] +a_{q+1} + a_p \& = a_{q+1} - a_p + a_{q+1} + a_p = 2a_{q+1},end{align}
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
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