General Investigations Regarding the Infinite Series (1812) by Carl Friedrich Gauss, translated from Latin by Wikisource
→
Carl Friedrich Gauss4463879General Investigations Regarding the Infinite Series 1812Wikisource
INTRODUCTION.
1.
The series which we propose to investigate in this treatise can be regarded as a function of four quantities which we shall call its elements. We will distinguish these by their order, with the first element being the second the third and the fourth It is clear that the first and second elements can be interchanged: therefore, if for the sake of brevity we denote our series by the symbol then we shall have
2.
By assigning definite values to the elements our series becomes a function of a single variable which is clearly cut off after the or term if or is a negative integer, but in other cases it extends indefinitely. In the former case, the series yields a rational algebraic function, but in the latter case, it usually yields a transcendental function. The third element must neither be a negative integer nor equal to zero, so that we do not have infinitely large terms.
3.
The coefficients of the powers in our series are as
and therefore they approach equality as the value of increases. So, if a definite value is also assigned to the fourth element the convergence or divergence of the series will depend on the nature of this value. Indeed, whenever a real value, positive or negative but less than unity, is assigned to the series, while not convergent immediately from the beginning, will nevertheless converge after a certain interval and will lead to a sum which is finite and determinate. The same will occur for an imaginary value of of the form whenever On the other hand, for a real value of greater than unity, or for an imaginary value of the form with the series will diverge, perhaps not immediately, but after a certain interval, so that it is meaningless to speak of its "sum". Finally, for the value (or more generally for a value of the form with ), the convergence or divergence of the series will depend on the nature of the elements as we will discuss, with particular attention to the sum of the series for in the third section.
It is therefore clear that, to the extent that our function is defined as the sum of a definite series, our investigation must, by its nature, be restricted to those cases where the series actually converges, and hence it is meaningless to ask for the value of the series for values of which are greater than unity. Furthermore, from the fourth section onwards, we will construct our function on the basis of a deeper principle, which permits the most general application.
4.
Differentiation of our series, considering only the fourth element as the variable, leads to a similar function, since it is clear that
The same applies to repeated differentiation.
5.
It will be worth our while to include here certain functions that can be reduced to our series and whose use is very common in analysis.
where the element is arbitrary.
where is an infinitely small quantity.
where is the base of the hyperbolic logarithm, and is an infinitely large number.
where are infinitely large numbers.
6.
The preceding functions are algebraic or transcendental depending upon logarithms and the circle. However, we do not undertake our general investigation for the sake of these functions, but rather to advance the theory of higher transcendental functions, of which our series encompasses a vast range. Among these, amid countless others, are the coefficients which arise in the expansion of the function into a series in terms of the cosines of the angles etc., about which we will speak particularly on another occasion. However, those coefficients can be reduced to the form of our series in several ways. Namely, setting
we have "firstly",
For if we view as the product of and (where denotes the quantity ), then is equal to the product of with
and
Since this must be identical to
the values given above are obtained automatically.
Secondly, we have
These values are easily derived from
Thirdly,
Finally fourthly,
These values and those following are easily derived from
FIRST SECTION. Relations between contiguous functions.
7.
We say that a function is contiguous with if it is obtained from the latter by increasing or decreasing the first, second, or third element by unity, with the remaining three elements being held constant. Thus the primary function produces six contiguous ones, any two of which are related to the primary function by a very simple linear equation. These equations, fifteen in number, are given below. For the sake of brevity we have omitted the fourth element, which is always understood to be and we have denoted the primary function simply by
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
8.
Now here is the proof of these formulas. If we set
then the coefficient of will be as follows:
in
in
in
in
Moreover, the coefficient of in , or the coefficient of in is
Hence, the truth of formulas 5 and 3 is immediately apparent. Formula 12 arises from 5 by swapping and and from these two, elimination yields 2. Similarly, by the same permutation, formula 6 arises from 3; combining 6 and 12 yields 9, permuting yields 14, and combining these gives 7. Finally, from 2 and 6, 1 is derived, and then by permutation, 10. Formula 8 can be derived in a similar manner to formulas 5 and 3 above, from the consideration of coefficients (if desired, all 15 formulas could be derived in a similar way), or more elegantly from the known equations, as follows.
By changing the element to and to in formula 5, we obtain
On the other hand, by changing only to in formula 9, we get
Subtracting these formulas immediately yields 8, and hence by permutation, 13. From 1 and 8, 4 follows, and then by permutating, 11. Finally, 15 is deduced from 8 and 9.
9.
If and are all integers (positive or negative), one can go from the function to the function , and likewise from there to the function through a series of similar functions, such that each one is contiguous to the preceding and succeeding ones. This is achieved by changing one element, e.g. , by one unit repeatedly, until one reaches and then changing the second element until one reaches and finally changing the third element until one reaches and so on to Since linear equations exist, according to art. 7, between the first, second, and third functions, and generally between any three consecutive functions in this series, it is easily understood that linear equations between the functions and so forth can be deduced by elimination. Thus, generally speaking, from two functions whose first three elements differ by integers, any other function with the same property can be obtained, provided that the fourth element remains the same. For what remains, it suffices to establish this remarkable truth generally; we shall not dwell on the shortcuts by which the operations required for this purpose can be made as brief as possible.
10.
Suppose that we are given e.g. the functions
between which a linear relation must be found. We can connect them through the following contiguous functions:
Thus we have five linear equations (from formulas 6, 13, 5 of art. 7):
Eliminating from I and II yields
Eliminating from this and III yields
Eliminating from IV and V yields
Finally, eliminating from this and VII yields
11.
If we wanted to exhaust all relations among triplets of functions where are either or then the number of formulas would increase to 325. Such a collection would not be useless; but it will suffice to present only a few here. These can be easily demonstrated, either from the formulas in art. 7 or, if one prefers, in the same manner as in art. 8.
etc., we obtain the following continued fraction for
[25]
where
etc., where the law of the progression is obvious.
Moreover, from equations 17, 18, 21, 22, we have
[26]
[27]
[28]
[29]
from which, substituting the values of the function into the continued fractions, an equal number of new continued fractions emerge.
Finally, it is clear that the continued fraction in formula 25 automatically terminates if any of the numbers is a negative integer, and otherwise it runs to infinity.
13.
The continued fractions in the previous article are of the utmost importance, and it can be asserted that hardly any continued fractions progressing according to a known law have so far been extracted by analysts, which are not special cases of ours. Especially notable is the case where we set in formula 25, so that and therefore, writing instead of
where
etc.
14.
It will be worth our while to include some special cases here. Setting it follows from formula I of art. 5 that
[31]
From formulas VI and VII of art. 5, we have
[32]
[33]
Changing the sign to here yields the continued fraction for
Furthermore, we have
[34]
[35]
Setting the continued fraction presented in art. 90 of Theoria motus corporum coelestium follows automatically from formula 30. Two other continued fractions are also proposed there, the development of which we thought to supply here. Setting
then (l.c.) hence
which is the first formula; the second is derived as follows. Setting
we have, by formula 25,
and
Hence
or by swapping the first and second elements,
However, by equation 21, we have
from which it follows that and substituting this value into the formula above yields
which is the second formula.
Setting in formula 30 yields, for an infinitely large value of
[36]
THIRD SECTION. On the sum of our series, with the fourth element set with a discussion of certain other transcendental functions.
15.
Whenever the elements are all positive quantities, all coefficients of powers of the fourth element become positive: and whenever one or another of those elements is negative, at least from some power onwards all coefficients will have the same sign, provided that is taken greater than the absolute value of the most negative element. It is clear from this that the sum of the series for cannot be finite unless the coefficients decrease to infinity after a certain term, or, to speak in the manner of analysts, unless the coefficient of the term is Indeed, for the benefit of those who favor the rigorous methods of the ancient geometers, we will show with all rigor that
first, the coefficients (since the series is not terminated) increase to infinity indefinitely whenever is a positive quantity.
second, the coefficients converge continually towards a finite limit whenever
third, the coefficients decrease to infinity indefinitely whenever is a negative quantity.
fourth, the sum of our series for notwithstanding convergence in the third case, is infinite whenever is a positive quantity or
fifth, the sum is truly finite whenever is a negative quantity.
16.
We will apply this general discussion to the infinite series etc., which is formed so that the quotients etc. resp. are the values of the fraction
for , etc. For brevity, we will denote the numerator of this fraction by and the denominator by . Furthermore, we assume that and are not identical, or equivalently that the differences etc., do not all vanish simultaneously.
I. Whenever the first of the differences etc., which does not vanish is positive, some limit can be assigned, beyond which the values of the functions and will always be positive and . It is evident that this occurs when is taken as the largest real root of the equation if this equation has no real roots at all, then this property holds for all values of . Therefore, in the series etc., at least after a certain interval (if not from the beginning), all terms will be positive and greater than unity. Consequently, if none of them tends to zero or infinity, it is clear that
the series etc., if not from the beginning, then at least after a certain interval, will have all its terms affected by the same sign and continually increasing.
By the same reasoning, if the first of the differences etc. which does not vanish is negative, then the series etc., will, if not from the beginning, then at least after a certain interval, have all its terms affected by the same sign and continually decreasing.
II. Now, if the coefficients are unequal, the terms of the series etc., will either increase or decrease to infinity, depending on whether the difference is positive or negative: we demonstrate this as follows. If is positive, let an integer be chosen so that and let , etc., and also Then it is clear that etc., are values of the fraction when , etc., while themselves are algebraic functions of the form
Therefore, since by hypothesis the difference is positive, the terms of the series etc. will, if not from the beginning, then after a certain interval, continually increase (by I). Hence the terms of the series etc., will necessarily increase beyond all limits, and therefore the terms of the series , etc., whose exponents are equal to , will do so as well. Q.E.D.
If is negative, then the integer must be chosen so that is greater than , and similar reasoning leads to the conclusion that the terms of the series
will continually decrease after a certain interval. Therefore, the terms of the series etc., and consequently also the terms of the series , etc., will necessarily tend to infinity. Q.E.S.
III. On the other hand, if the coefficients are equal, then the terms of the series etc., converge continually to a finite limit: we demonstrate this as follows. First, let us suppose that the terms of the series increase continually after a certain interval, or equivalently that the first of the differences etc. which does not vanish is positive. Let be an integer such that becomes a positive quantity. Set
and such that , etc., are values of the fraction when , etc. Therefore, since we have
and since is a negative quantity by hypothesis, the terms of the series etc., will decrease continually after a certain interval. Therefore, the corresponding terms of the series etc. which are always smaller, while also increasing continually, must converge to a finite limit. Q.E.D.
If the terms of the series etc., decrease continually after a certain interval, an integer must be chosen such that becomes a positive quantity. It then becomes evident from entirely similar reasoning that the terms of the series
increase continually after a certain interval. Therefore, the corresponding terms of the series etc., which are always greater, while also decreasing continually, must converge to a finite limit. Q.E.S.
IV. Lastly, concerning the sum of the series whose terms are etc. in the case where these terms decrease indefinitely, let us first suppose that falls between and , meaning that is either a positive quantity or Let be a positive integer, chosen arbitrarily in the case where is positive, or so that it makes the quantity positive in the case where Then we will have
where either is positive, or, if it equals then at least will be positive. Hence (by I), a value can be assigned to the quantity , which, once exceeded, will ensure that the values of the fraction will always be positive and greater than unity. Let be an integer greater than and also greater than and let the terms of the series etc., corresponding to the values etc., be denoted by etc. Then
will be positive quantities greater than one, so that
Consequently, the sum of the series will be greater than the sum of the series
no matter how many terms are included. However, as the number of terms increases indefinitely, the latter series exceeds all limits, as the sum of the series is known to be infinite and remains infinite even if the terms are removed from the beginning. Hence, the sum of the series and consequently the sum of of which it is a part, increases beyond all limits.
V. However, when is a negative quantity that is absolutely greater than one, the sum of the series will certainly be finite when continued indefinitely. Indeed, let be a positive quantity less than Then similar reasoning shows that the quantity can be assigned a value beyond which the fraction always has positive values less than unity. Now, if we take an integer greater than and let the terms of the series etc., corresponding to the values etc., be denoted by etc., then
Consequently, the sum of the series no matter how many terms are included, is less than the product of with the sum of the same number of terms of the series
However, the sum of this series can be easily found for any number of terms. In particular,
The first term
The sum of the first two terms
The sum of the first three terms
etc.
and since the second part (by II) forms a series which decreases beyond all limits, the sum must be Hence when continued infinitely, will always remain less than and thus will certainly converge to a finite sum. Q.E.D.
VI. In order to apply those general assertions concerning the series etc. to the coefficients of the powers etc. in the series it is necessary to set from which the five assertions in the preceding article follow automatically.
17.
Therefore, investigations of the nature of the sum of the series are naturally restricted to the case where is a positive quantity, in which case the sum will always be a finite quantity. However, we must begin with the following observation. If, after a certain term, the coefficients of the series decrease beyond all limits, then the product
must when for even if the sum of the series becomes infinitely large. For since the sum of two terms is the sum of three is the sum of four is , etc., the limit of the sum when continued indefinitely will be Therefore, whenever is a positive quantity, we must have for and hence, by equation 15 of art. 7,
or
[37]
Thus, similarly, we have
and so on, where denotes an arbitrary positive integer,
the product of
with
and
divided by the product of
with
18.
We now introduce the notation
[38]
where is naturally restricted to be a positive integer, and with this restriction, represents a function determined solely by the two quantities and . Then it is easy to understand that the theorem proposed at the end of the preceding article can be expressed as follows:
[39]
19.
It will be worthwhile to examine the nature of the function in more detail. Whenever is a negative integer, the function evidently has an infinitely large value, as long as a sufficiently large value is assigned to . For non-negative integer values of , we have:
etc., and generally:
[40]
For arbitrary values of , we have:
[41]
[42]
and therefore, since
[43]
20.
It is worth giving special attention to the limit toward which the function \Pi(k, z) continually converges, as k increases to infinity. First, let be a finite value of which is greater than . Then it is clear that, as increases from to the logarithm of receives an increment which can be expressed by the following convergent series:
Therefore, as increases from to the logarithm of will receive an increment
which will always remain finite, even when tends to infinity, as can be easily demonstrated. Therefore, unless an infinite factor is already present in i.e., unless is a negative integer, the limit of as tends to infinity will certainly be a finite quantity. Hence, it is evident that depends solely on or in other words, it is a function of alone, which we will simply denote by We therefore define the function as the value of the product:
for or, if one prefers, as the limit of the infinite product
21.
Immediately following from equation 41, we have the fundamental equation:
[44]
Hence, in general, for any positive integer
[45]
For a negative integer value of the value of the function will be infinitely large; for non-negative integer values, we have
and, in general
[46]
However, this property of our function should not be mistaken as its definition, as it is inherently limited to integer values and there exist countless other functions (e.g., , etc., where denotes the circumference of a circle of radius ) that share the same property.
22.
Although the function may appear to be more general than it will henceforth be redundant for us, as it can easily be reduced to the latter. Indeed, it follows from the combination of equations 38, 45, and 46 that
[47]
Moreover, the connection of these functions with that which Kramp has called "facultates numericae" is evident. Specifically, the facultates numericae, which this author denotes by , can be expressed in our notation as:
However, it seems more advisable to introduce a function of one variable into the analysis, rather than a function of three variables, especially when the latter can be reduced to the former.
23.
The continuity of the function is interrupted whenever its value becomes infinitely large, i.e., for negative integer values of Therefore, it will be positive from