Abel's Test - Abel's Test in Complex Analysis

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

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

and the series

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

converges when |z| < 1 and diverges when |z| > 1, and the coefficients {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

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

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

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

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

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

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(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.

Famous quotes containing the words test, complex and/or analysis:

“The way of paradoxes is the way of truth. To test Reality we must see it on the tight-rope. When the Verities become acrobats we can judge them.”
—Oscar Wilde (1854–1900)

“The human mind is so complex and things are so tangled up with each other that, to explain a blade of straw, one would have to take to pieces an entire universe.... A definition is a sack of flour compressed into a thimble.”
—Rémy De Gourmont (1858–1915)

“... the big courageous acts of life are those one never hears of and only suspects from having been through like experience. It takes real courage to do battle in the unspectacular task. We always listen for the applause of our co-workers. He is courageous who plods on, unlettered and unknown.... In the last analysis it is this courage, developing between man and his limitations, that brings success.”
—Alice Foote MacDougall (1867–1945)