Volume 1 / Part 5 — *230. ON CONVERGENTS.
Reading edition
*230. ON CONVERGENTS.
Summary of *230.
In the present number, we have to consider the notion of a function converging into a given class, or, as we may express it, the notion that the value of the function "ultimately" belongs to the given class. If is the function in question, the given class, and a series to which the arguments belong, we say that " is -convergent into " if there is an argument such that, for all arguments from onward (in the -order), the value of the function is an . That is, is -convergent into if A term which is of this nature is said to belong to the class . Thus is -convergent into if the class is not null. Hence we have the following pair of definitions:
In all the cases that have any importance, will be a one-valued function (i.e. a one-many relation), will be a series, and will be a class having no maximum in . For, if has a maximum in , then the classes into which converges are simply those to which the value for this maximum belongs. The following propositions, though only important under the above circumstances, are in general true under much wider hypotheses.
It is possible to generalize still further the notion of convergence, so as to apply to any property which belongs to when confined to sufficiently late arguments. For this purpose, we have to consider where is to be confined to terms later than or equal to some term . If, under these circumstances, always belongs to the class , we may say that ultimately becomes a . We may put
This is the general conception of which is a particular case; in fact, will have to be used when the ultimate properties of the function with which we are concerned are not properties of its values; but when they are properties of its values, enables us to deal with them more easily than .
In this number, we prove the following propositions among others:
*230·171.
*230·211.
*230·253.
*230·4.
*230·42.
*230·53.
In virtue of this proposition, the case when is uninteresting, and in order to obtain interesting interpretations of our propositions, it is necessary to suppose that has no maximum. Similarly when, in later numbers, we consider , we shall only obtain interesting results when this has no maximum, which requires that should be a compact series and should be dense in . These assumptions are, however, not usually required for the truth of our propositions.
*230·01.
*230·02.
*230·1.
*230·11.
*230·12.
Dem.
*230·13.
Dem.
*230·131.
*230·14.
Dem.
*230·141.
*230·142.
*230·15.
*230·151.
*230·152.
*230·16.
Dem.
*230·161.
*230·17.
Dem.
*230·171.
Dem.
*230·21.
*230·211.
*230·22.
*230·221.
*230·23.
Dem.
*230·231.
*230·24.
*230·25.
Dem.
*230·251.
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*230·252.
*230·253.
*230·31.
Dem.
*230·311.
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*230·32.
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*230·321.
Dem.
*230·4.
Dem.
*230·41.
Dem.
*230·42.
Dem.
*230·421.
*230·51.
Dem.
*230·511.
Dem.
*230·512.
Dem.
*230·513.
Dem.
*230·514.
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*230·52.
Dem.
*230·53.
Dem.
*230·54.
Dem.