Infinite series

“ An "infinite series" is therefore a contradiction in adjecto. "As every number, although immeasurably and inconceivably great, is impossible without unity as its basis, so every series, being itself a number, is impossible unless a first term is given as its commencement.... Even if it should be allowed that the series has no first term, but has originated ab æterno, it must always at each instant have a last term; the series as a whole can not be infinite." [117] If one thing more can be added to the number of existing things in the universe, then it is not infinite in number or in extent. ”

“ The great mathematicians of the 18th century used infinite series freely with very little regard to their convergence or divergence and with, occasionally, very extraordinary results. Series which are ultimately divergent may be used to calculate values of functions in special cases and to represent what are called “asymptotic expansions" of functions (see Function) .
Infinite Products.
21. The product of an infinite number of factors formed in succession according to any given law is called an infinite product. The infinite product Πn≡ (1 +u1) (1+u2) . ”

“ For let us assume that the world has no beginning as regards time, so that up to every given instant an eternity has elapsed, and therefore an infinite series of successive states of the things in the world has passed by. But the infinity of a series consists just in this, that it can never be completed by successive synthesis. Therefore an infinite past world-series is impossible, and accordingly a beginning of the world is a necessary condition of its existence ”

“ Here is zero or unity according as is even or odd, and although does not become infinite as increases without limit, it does not tend to a limit, but oscillates. It is evident that if all the terms of a series have the same sign, the series cannot oscillate.
Since the sum of a converging series is a perfectly definite number, while such a thing as the sum of a nonconvergent series does not exist, it follows at once that it is absolutely essential in any given problem involving infinite series to determine whether or not the series is convergent. ”

“ The sphere of change, however, one would answer, includes all nature, and science in its discoveries acts on the hypothesis that these steps of change may be infinitely divisible. Royce held to it firmly that any consistent attempt to make an orderly arrangement of the terms of an infinite whole must lead to the Indefinite Regress. And he further shows the connection with the fact that an infinite series can be adequately represented by a part of itself. ”

“ Time, therefore, as the evolution of that activity, cannot be produced by composition. Now, as all other infinite series are only imitations of the originally infinite series, Time, no infinite series can be otherwise than continuous. ”

“ That the mathematician cannot actually present before us the whole of an infinite series, is indeed most certain; for such, power belongs only to an Infinite Being. But does the fact that man’s faculties are limited, prove that an arithmetical process cannot be carried on from eternity to eternity? ”

“ This absolute productivity must pass over into an empirical nature. In the idea of absolute productivity, is the thought of an ideal infinity. The ideal infinity must become an empirical one.
But empirical infinity is an infinite becoming. Every infinite series is but the exhibition of an intellectual or ideal infinity. The original infinite series (the ideal of all infinite series) is that wherein our intellectual infinity evolves itself, viz., Time. The activity which sustains this series is the same as that which sustains our consciousness; consciousness, however, is continuous. ”

“ None as long as you deal with only finite quantities. The absurdity is in the fact that the infinite divisibility of matter implies an infinity of parts; and an infinity of parts, an infinity of numbers; and numbers and every series of numbers may be increased by addition, and diminished by subtraction. An infinite series is impossible.
The moment the scientists leave the domain of particulars or positive facts, and attempt to induce from them a law, their induction is of no value. ”

“ An infinite series may contain both positive and negative terms. The terms may be positive and negative alternately or they may occur in groups which without altering the order of the terms of the series may each be collected into a single term; thus all series may be regarded as belonging to one of two types, ul-i-u2+u3+ in which the terms are all positive, or u1-u2-|-u3- in which the terms are alternately positive and negative. 8. It is clear that if a series is convergent un must tend to the limit zero as n is increased indefinitely. ”

“ When the cardinal number of terms in the field of a series is infinite, the relation-number of the series is not determined merely by the cardinal number, indeed an infinite number of relation-numbers exist for one infinite cardinal number, as we shall see when we come to consider infinite series. ”

“ It may well be that within our experience there is nothing which will serve. But let us suppose that that which in material terms we represent as an infinite series is a will—a will in contact with all existence, as shown by the properties it had when we conceived it as an ultimate medium. For, regarding it as an ultimate substance, we found that it would be affected by pulsations infinitely quicker than light and electricity ”

“ The sum of an infinite series may be deduced from the sum to rt terms, when this is known, by increasing n indefinitely and finding the limit, if any, to which it tends, but a series may often be summed to infinity when it cannot be summed to n terms ”

“ He was the first to observe rigour in the treatment of infinite series, the first to fully recognise and emphasise the importance, and to make systematic use of determinants and of imaginaries, the first to arrive at the method of least squares, the first to observe the double periodicity of elliptic functions. ”

“ In Case II the infinite series is said to be nonconvergent [3] . Series under this head may be divided into two classes.
First Class. Divergent series, in which the sum of terms increases indefinitely in numerical value as increases without limit; take for example the series
. As increases without limit, increases without limit and therefore the series is divergent.
Second Class. Oscillating series, of which
is an example. ”

“ He warns his readers occasionally against the use of divergent series, but is nevertheless very careless himself. The rigid treatment to which infinite series are subjected now was then undreamed of. No clear notions existed as to what constitutes a convergent series. ”

“ But it must not be supposed that all infinite numbers are equal. On the contrary, there are infinitely more infinite numbers than finite ones. There are more ways of arranging the finite numbers in different types of series than there are finite numbers. There are probably more points in space and more moments in time than there are finite numbers. There are exactly as many fractions as whole numbers, although there are an infinite number of fractions between any two whole numbers. But there are more irrational numbers than there are whole numbers or fractions. ”

“ The various series of causes and effects are partly contingent relatively to each other, and are partly themselves continued into the so-called Infinite, and thus contain in their content nothing but those situations and forms of existence of which each is finite in itself; and what ought to give stability to the connection of the series, the Infinite namely, is not only something above and beyond this world, but is a mere negative, the very meaning of which is relative merely, and is conditioned by what is to be negated by it, and is consequently for this very reason not negated. ”

“ But the mind perceives things as they exist; and the line is carried on in idea without end. In that respect an unbounded prospect is similar to an infinite series. In fact, the uneasiness of an unbounded prospect differs very little in its feeling from that of an infinite series ”

“ The supposed difficulties of continuity all have their source in the fact that a continuous series must have an infinite number of terms, and are in fact difficulties concerning infinity. Hence, in freeing the infinite from contradiction, we are at the same time showing the logical possibility of continuity as assumed in science. ”

“ This theorem provides a means for computing the values of a function at points near to an assigned point when the value of the function and its differential coefficients at the assigned point are known. The function is expressed by a terminated series, and, when the remainder tends to zero as n Taylor’s Theorem. increases, it may be transformed into an infinite series. ”

“ When we trace backwards a series of finite yet intelligent beings, we must arrive at one of two conclusions:—We must either trace the series through an infinite and eternal succession of finite beings, each the cause of the one which succeeded it;—or we must refer the com [Pg 165] mencement of the series to one great intelligent being, himself uncaused, infinite, and eternal. To trace the series to one being, finite, yet uncaused, is totally inadmissible; and not less so is the conception of finite beings in an infinite and eternal series. ”

“ If then such infinites cannot be inherent in the one, neither will there be infinites in ascending series. Still it is necessary that all should be inherent in the first, for example, in number, and number in them, so that they will reciprocate, but not be more widely extensive. Neither are those infinite which are inherent in the definition of a thing, for if they were, we could not define, so that if all predicates are predicated per se, and these are not infinite, things in an upward progression will stop, wherefore also those which descend. ”

“ It is an idea which implies contradiction; and all these ‘infinites’ and ‘infinitesimals’ are of use only in geometry, as imaginary roots are in algebra.” That which is ordinarily called the infinite, that is, the quantitative infinite, is in reality only the indefinite. “We involve ourselves in difficulty when we talk about a series of numbers extending to infinity; we imagine a last term, an infinite number, or one infinitely little. ”

“ As regards the virtual and the accidental existence of the infinite, it has been established in some cases; it has been proved, e.g., that a body can virtually be divided ad infinitum, also that time can be divided ad infinitum; in other cases it is still an open question, as, e.g., the existence of the infinite in succession, which is called the accidental infinite, i.e., a series of things in which one thing comes forth when the other is gone, and this again in its turn succeeded a thing which had ceased to exist, and so on ad infinitum. ”

“ It is obvious that all the members of any series of causes essentially subordinate the one to the other must exist simultaneously. Whether such a series could be infinite depends, therefore, on the question whether an actually infinite multitude is intrinsically possible. ”

“ And inasmuch as between every one of these combinations and its next recurrence, every other possible combination would necessarily have been undergone, and since every one of these combinations would determine the whole series in the same order, a circular movement of absolutely identical series is thus demonstrated; the Universe is thus shown to be a circular movement which has already repeated itself an infinite number of times, and which plays its game for all eternity. ”

“ To ensure infinite divisibility, we require also that every event should enclose events other than itself. Assuming these characteristics, temporal enclosure is an infinitely divisible point-producer. We can now form an “enclosure-series” of events, by choosing a group of events such that of any two there is one which encloses the other; this will be a “punctual enclosure-series” if, given any other enclosure-series such that every member of our first series encloses some member of our second, then every member of our second series encloses some member of our first. ”

“ Although there are but two grades of magnitudes of infinite collections, yet when certain conditions are imposed upon the order in which individuals are taken, distinctions of magnitude arise from that cause. Thus, if a simply endless series be doubled by separating each unit into two parts, the successive first parts and also the second parts being taken in the same order as the units from which they are derived, this double endless series will, so long as it is taken in that order, appear as twice as large as the original series. ”

“ Let us try and appreciate the fact that, under our present conditions, our conceptions of the immense and minute—namely, extension in Space, and that of quick and slow or duration in Time—are purely relative, and that from this arise those pseudo-conceptions which we call the infinitely extended and the infinitely lasting. Under our present limitations it is impossible for us to grasp the whole of any Truth, if we could do that, there would be no such mystery of Infinity to puzzle us ”

“ Our different conceptions of the infinity of number contrasted with those of duration and expansion. It will, perhaps, give us a little further light into the idea we have of infinity, and discover to us, that it is nothing but the infinity of number applied to determinate parts, of which we have in our minds the distinct ideas, if we consider that number is not generally thought by us infinite, whereas duration and extension are apt to be so ”