Ratio test

The ratio test is a mathematical convergence criterion for infinite series. It is based on the comparison test, that is, a complicated series is estimated by a simple, here the geometric series, upward. The geometric series converges, when the amount of the additional elements is decreased, that is, the (constant ) the quotient of two successive links q is less than 1. If a different number from a certain element at least as quickly, so the quotient is less than or equal to q, this shall also be convergent. With the quotient criterion divergence can be detected. The quotient is always greater than or equal to 1, the amount of the additional elements is not smaller. Since then not form a null sequence, the series is divergent.

Developed the quotient criteria of the mathematician and physicist Jean -Baptiste le Rond d' Alembert, in whose honor this mathematical statement is also known as d' Alembert's convergence criterion.

Statement

Given an infinite series with real or complex summands, for almost all. Is there a such that for almost all applies

So the series is absolutely convergent. Other hand, applies to almost all

Then the series is divergent.

In the case of convergence must be independent of and strictly less than 1. Applies only, so can approach arbitrarily close to 1, so the ratio test does not provide information on the convergence or divergence.

Examples

Example 1 We consider the series

And check them for convergence. Will give us both the ratio test:

Consequently, the series is convergent.

Example 2 We consider the series

And check them for convergence. We obtain:

Thus, this series is divergent.

Example 3 Let the radius of convergence of the power series

Determine for complex numbers. For the series is obviously convergent to 0, so be and we get

Thus, the series has radius of convergence.

Idea of ​​proof

The case of the convergence follows by the comparison test the convergence of, a geometric series. The criterion for divergence, it follows that the links can then form because no zero sequence.

An example of the non-applicability of the quotient is the general criterion harmonic series. It is

For the general harmonic series is divergent, convergent for; but the ratio test can not distinguish the two cases.

Special cases

Exist, then provides the ratio test

Using limit superior and limit inferior can be the ratio test formulated as follows:

In contrast to the root criterion must be used for the inferior divergence criterion is not the limit superior, but the limes.

Modified ratio test

In addition to the "normal " ratio test, there are the following versions (see also criterion of Raabe ): Let a sequence of real positive terms. now, if

So true that is convergent.

On the other hand

As follows:

Applications

The ratio test can be, for example, the convergence of the Taylor series for the exponential function and the sine and cosine functions show.