# 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, and , if the sequences satisfy

- converges

- is monotonic and

then the series

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

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*.^{[1]}

**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(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

## Notes

- ↑ (Moretti, 1964, p. 91)

## References

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

de:Kriterium von Abel sv:Abels sats