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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, $\left\{a_n\right\}$ and $\left\{b_n\right\}$, if the sequences satisfy

* $sum^\left\{infty\right\}_\left\{n=1\right\}a_n$ converges

* $lbrace b_n rbrace,$ is monotonic and $lim_\left\{n rightarrow infty\right\} b_n ne infty$

then the series

$sum^\left\{infty\right\}_\left\{n=1\right\}a_n b_n$

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_{nrightarrowinfty} 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 {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.

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


z = e^{itheta} quadRightarrowquad z^{frac{1}{2}} - z^{-frac{1}{2}} = 2isin{textstyle frac{theta}{2}} ne 0

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


begin{align} 2isin{textstyle frac{theta}{2}}left(S_p - S_qright) & = 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_nright) z^{n-frac{1}{2}}right] - a_{q+1}z^{q+frac{1}{2}} + a_pz^{p+frac{1}{2}}, end{align}

where Sp and Sq are partial sums:


S_p = sum_{n=0}^p a_nz^n.,

But now, since |z| = 1 and the an are monotonically decreasing positive real numbers when n > m, we can also write


begin{align} left| 2isin{textstyle frac{theta}{2}}left(S_p - S_qright)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_nright) 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_nright)right] +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 aq+1 can be made smaller than any given ε > 0 by choosing a large enough q.