Positive series

Mathematical terminology

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The positive term series is a mathematical term. In the series theory, the positive term series is a very important kind of series. The research on general series can sometimes obtain results through the research on positive term series, just like the relationship between the generalized integral of non negative function and the general generalized integral. The so-called positive term series is such a series: each term of the series is non negative. The methods for judging the convergence of positive term series mainly include: using partial sum sequence, comparison principle, comparison method, radical method, integral method and Rabe method.

Chinese name: Positive series
Foreign name: Positive series
Definition: Each item is a series composed of positive numbers

Important series: P series
Convergence criterion: Comparison principle, comparison discrimination, radical discrimination
Applied discipline: mathematical analysis

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definition

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If the symbols of several series are the same, they are called series with the same sign. For the series with the same sign, we only need to study the series that all items are composed of positive numbers, and call it the positive term series. If all the terms of the series are negative, then it will be multiplied by - 1 to get a positive term series. They have the same convergence and divergence.^[1]

In other words, if

, then series

by Positive series 。^[2]

Convergence criterion

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Partial sum sequence discrimination

Partial sum sequence of positive series

Is monotonically increasing series That is:

，

The necessary and sufficient condition for convergence is bounded, so there are:

Positive series

The necessary and sufficient condition for convergence is: its partial sum sequence

Bounded, that is, there is some positive number

, for all positive integers

yes

。^[1]

Principle of comparison

set up

and

Is a series of two positive terms, if there is a positive number

, so that everything

All have

, there are:

(1) If series

Convergence, then series

also convergence ；

(2) If series

Divergence, then series

also Divergence 。^[1]

Comparison discriminant（ D'Alembert discrimination ）

set up

It is a positive series, and there is a normal number

And constant

。

(1) If all

, establish inequality

, then the number of stages

Convergence;

(2) If all

, establish inequality

, then the number of stages

Divergence.

Limit form of comparison discrimination:

set up

Is a positive series, and

, there are:

(1) When

Time, number of stages

Convergence;

(2) When

Time, number of stages

Divergence.

Note: If

At this time, the convergence and divergence of the series cannot be judged by the comparison method, because it may be convergent or divergent, such as the series

and

, their ratio limit is

, but

Is convergent,

It is divergent.^[1]

Radical discriminant method (Cauchy discriminant method)

set up

It is a positive series, and there is a normal number

And normal number

。

(1) If all

, establish inequality

, then the number of stages

Convergence;

(2) If all

, establish inequality

, then the number of stages

Divergence;

Limit form of Cauchy discriminant method:

set up

Is a positive series, and

, then:

(1) When

Time, number of stages

Convergence;

(2) When

, Series

Divergence.

Note: If

At this time, the radical method cannot be used to judge the convergence and divergence of a series, because it may be convergent or divergent, such as a series

and

, their ratio limit is

, but

Is convergent,

It is divergent.^[1]

Integral discriminant method

The integral discrimination method uses the monotonicity and integral properties of non negative functions, and takes Anomalous integral To judge the convergence and divergence of positive series for comparison purposes.

set up