Root test

The root test ( Cauchy ) ( after the French mathematician Augustin Louis Cauchy [ 1789-1857 ] ) is a mathematical convergence criterion for infinite series. It is based, as the ratio test on a comparison with a geometric series.

The basic idea is this: A geometric series with real and positive elements converges if and only if the quotient q successive links is less than 1. The nth root of the nth summand of this geometric series tends to q. Behaves another series as well, it too is convergent. Since they are well to absolute convergence, the rule can be generalized by considering the amounts.

Formulations

Be given an infinite series with real or complex summands. If you now

Can prove, so the series is convergent. Then converges even absolutely, that is, the series also converges.

However, if

So the series diverges because the terms of the series do not form a null sequence.

In the case

Can be nothing about the convergence of the series testify. Thus, for example, by the root test no statement about the convergence of the general harmonic series for it, since

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

Examples

Example 1 We examine the number

On convergence. We receive on the root test:

With the Euler's number. Thus, this series is convergent.

Example 2 We now consider the series

On convergence. We obtain:

Thus, this series is divergent.

Sketch of proof

The root criterion was first proved by Augustin Louis Cauchy. It follows with the majorant criterion Properties of the geometric series:

• Because, for all, the majorant criterion is met with a convergent geometric series as majorant.

• The situation will change nothing, if this criterion is not met for the first N terms of the series.

• Applies, as is true for almost all n, by definition, the largest accumulation point, bringing back a majorant can be constructed.

Remainder estimate

If the number after the root test convergent, one still gets an error estimate, ie an estimate of the remainder term of the sum by N summands:

The root criterion is sharper than the ratio test

Be a positive result and was

Narrated in a row the

Ratio test, a decision ( ie, in the case of convergence, or in the case of divergence)

So also the root test provides a decision ( ie, in the case of convergence, or in the case of divergence).

This is induced by the chain of inequalities

Is without limitation, and so there is every little () but positive index barrier from the true

Multiplying the inequality by up we get in the middle of a telescopic product.

Multiplying with on and pulls the nth root is then so

For converges to the left side and to the right side. is therefore

Since can be chosen arbitrarily small, therefore follows

For example, if the terms of the series and then and.

Here is and what the ratio test does not provide a decision.

The root test provides a decision here, but because it is.

It follows from the convergence of. The root test is so really sharper than the ratio test.

Swell

• Convergence criterion