Radius of convergence

The radius of convergence is in the Analysis a property of a power series of the form

Indicating what area of the complex plane for the power series absolute convergence is guaranteed.

Definition

The convergence radius is defined as the supremum of all numbers defined by the power series converges for all for which:

If the power series converges for all real numbers or on the whole complex plane, they say, the radius of convergence is infinite.

If the radius of convergence, then shows that the power series for all with diverges.

If a real power series considered, whose coefficients are real numbers, and are also real, then the region of convergence for the resolution of Betragsungleichung the interval as well as possibly one or both of the boundary points. For power series in the complex, that is, all of these variables can be complex numbers, is the region of convergence of series of functions from inside the disc around the center and radius of the circle of convergence, and possibly from some of its boundary points.

Determination of the radius of convergence

The radius of convergence can be calculated with the formula of Cauchy -Hadamard: it is

This applies if the limit is superior in the denominator the same, and if it is the same.

If at a certain index of 0 are all different and the following limit exists, then the radius of convergence can be easier by

Be calculated. But this formula is not always applicable, for example, in the sequence of coefficients: The associated series has radius of convergence 1, but the specified limit does not exist. The formula of Cauchy -Hadamard, however, is always applicable.

Consequences of the radius of convergence:

• If so, the power series is absolutely convergent.
• If so, the power series is divergent.
• If so, no general statement can be made, in some situations, but it helps the Abelian limit theorem.
• If so, the power series converges uniformly for all x. On an inner circle or sub-interval is therefore always before a uniform convergence.

Examples of different boundary behavior

The following three examples each have radius of convergence 1:

• Converges at any of the boundary points.
• Converges in the real domain at both boundary points.
• Converges in the real not the " top " edge point ( harmonic series ), but rather on the " bottom " edge point ( alternating harmonic series ).

Influence the development point on the radius of convergence

The development point of a power series has a direct influence on the coefficient sequence and thus also on the radius of convergence. For example, considering the analytical function

In its power series representation

These transformations follow directly by means of the geometric series. This representation corresponds to the power series around the expansion point and the root test follows the radius of convergence.

If one chooses, however, as a development point, it follows with some algebraic transformations

Here follows by means of the root criteria on the convergence radius.

A third point of development provides with similar procedure

As a power series representation with the radius of convergence. If one draws these three radii of convergence to their development points, they all intersect in the point since the function f has a singularity and is not defined. Clearly therefore expands the circle of convergence to a development point until it hits an undefined point of the function.

Derivation

The formulas for the radius of convergence can be derived from the convergence criteria for rows.

Root test

The formula of Cauchy -Hadamard is derived from the root test. By this criterion, the power series converges

Absolutely when

Solving for gives the radius of convergence

Ratio test

If almost all of which are non-zero, the power series is converged by the ratio criterion, where the following condition is satisfied:

Solving for yields:

Thus, the power series converges for. This is generally not the radius of convergence. The reason is that the ratio test in the following sense is strictly slower than the root test: Is

So can not be concluded in general that the series diverges. The divergence is obtained from but

Similar as above, so it is concluded that the power series diverges for, where

One can therefore only generally state that the radius of convergence is between and.

But it follows in particular: From the existence of this and in this particular case

Submitted radius of convergence.