SECTION 45.

The practical addition, subtraction, and multiplication of two such infinite series R and S, proceeding according to whole powers of x, that is, the transformation of the sum R+S, the difference RS, and the product R. S into similar infinite series, presents, according to what has preceded, no difficulty at all.

Ꭱ

But if the quotient ૩. ', or g

S'

or R.

S

has to be transformed

...

into a similar infinite series, then the coefficients A., A1, A ̧, A., &c. &c. of an infinite series A+ A1. x + А ̧.x2 + А ̧.x3 + have to be sought, such that the latter when multiplied by S gives R or 1; now the multiplication and comparison of the product with the series R or 1, goes on in infinitum; and we obtain new equations in infinitum between the coefficients, so that from each consecutive equation a consecutive coefficient is continually found from among the coefficients A., A1, A2, A., &c. which were all at first unknown, and this in infinitum. The new infinite series A+ A1 x + А ̧x2 + А ̧ï3 +... has therefore in infinitum the property, that when multiplied by S, the result is a new series which coincides in infinitum with the series R or with 1. Now since the quotient or the quotient is by (sect. 18) a mere form, which represents the property (and therefore every expression which possesses this property) that

R

R

S

1

1

S

S when multiplied by S gives R,-or that multiplied by S

S

gives 1, and since the infinite series A+ A1x+Аx2+... just found possesses this property, it follows by (sect. 3) that it is

66

R 1

equal" to the quotient or S S' and may be every where

R 1

substituted in "calculation" for the quotient or

S S

uncon

ditionally, without our having to fear that we shall thereby come into contradiction with the laws of operation *.

whereas it ought to hold for all values of x, such a conclusion might be permitted. But then we have also =0,

that is, this last equation is only necessarily correct when x is not zero, since for x=0 the equation (2) is already satisfied, without requiring the second factor а 1 + a2 x +α 2 x2+... to be zero. That therefore a1 =0 necessarily does not follow, because the equation (3) does not necessarily hold for x=0.

But if we keep the view given in the four preceding chapters carefully in mind,

the explanation furnished in the text presents no difficulty whatever.

* We obtain the same result if we apply the formula (sect. 17, No. 10)

We may add here, by way of illustration,

(1) The coefficients A。, A1, A,, A., &c. cannot be determined when more first coefficients in the divisor S are equal to zero than in the dividend R*, in other words, when the

m

x2 + Cn+3 • x3 +

while m<n, and b and c, are not equal to zero. This is also perfectly self-evident, because the result of the transformation can be at most 1 (Co+ C1. x + C2. x2 + С3. x3 +

n-m

.).

(2) If it has been shewn that the infinite series

A+ A1. x + A ̧· x2 + А ̧· x3 + ...

R
S

is "equal" to the quotient it follows as a matter of course that

this is no longer the case for any finite number of terms of this series, however many we may take of them. Hence

if we only wished to retain a number n of the first terms of the infinite series thus found, we should be obliged to subjoin a complementary term, i.e. we must conceive an expression E, which is usually unknown, and cannot perhaps be exhibited in a finite form, to be subjoined, so that

R

S'

shall be again "equal" to the quotient This complementary term E is (although not in the above case, yet in the following developments) usually nothing but a symbol, which represents the sum of all the following infinitely many terms of the infinite series first found, and therefore itself denotes an infinite series.

(3) Since the coefficients of the infinite series R and S are arbitrary, they may, beginning from a certain term, be conceived as all equal to zero in infinitum. In this case we have transformed a broken rational function of x, as

continually in infinitum, and at the same time do not forget that A and B must here be treated as whole functions of x. If we were to stop in this division, we should have a complementary term in addition to the finite number of terms hitherto found, in order to obtain a correct equation; but if we conceive the division to be really continued in infinitum, no complementary term can ever appear.

* The general calculation would namely in this case give the form for Ao, which form does not shew that the coefficient is infinitely great, but which form always shews that the general method of calculation does not apply to this particular case, and that this case must be particularly treated.

into an infinite series which may be every where substituted for it without having to fear a contradiction on the part of the laws of operation.

R- -m

=

1

Rm

SECTION 46.

Since when m is a positive whole number, the mth power of a series proceeding according to powers of x, is nothing but the product R.R.R... of m factors, there is no difficulty in the transformation of Rm, when m is a positive whole number, into a similar series. The same is also the case with R-", since and the division has been completed in (sect. 45). On the other hand we cannot at present speak of a broken power of an infinite series, since we at present are only acquainted with such broken powers as have positive dignands (comp. sect. 28), while such a series as R is here considered as a general form, in which we have not to regard the signification of the several letters.

But if we have to extract the mth root of the series R proceeding according to whole powers of x, i. e. if we have to transform /R into a similar series, we must find the coefficients Ao, A1, A2, A3, &c. of an infinite series

A+A,.x + Ag. x2 + A3. x3+...

so that when this series is potentiated by the positive whole number m, the series R results. Now if we really raise this series

A。+ A1.x + A. x2 + ...

to the mth power, and compare the result with the series R, we obtain, since the coefficients of like powers of a must be equal, an infinite number of equations between these coefficients; of which each one in succession in infinitum will determine a consecutive one of the coefficients A, A1, A2, A3, &c. &c. which were at first unknown. Therefore there exists an infinite series, A+ A1a+A2+... which when potentiated by m, gives a new infinite series, whose terms coincide in infinitum with those of the series R. Now since "/R represents each of m expressions, which have the property of, when potentiated by m, giving the radicand R, (by sect. 41), and since the infinite series A2+A1x + A+ A+... just determined has this property, we have the generally correct equation (sect. 3)

"/R = "/1. (A。+ A1. x + А2 . x2 + A3 ⋅ x3+ .),

2

in case the coefficients A, A1, A2, &c. &c. have been so chosen as to be only single-meaning for the more convenient discovery of the corresponding values.

We find, however, in actually performing the process, this exception: the coefficients A, A, A, &c. can not be determined when a certain number, which is not exactly m, or 2m, or gene

rally vm, of the first coefficients of R are successively = 0*. But this is self-evident in a direct method, because the series R can then be reduced to the form

th

x”. (b。 + b2 x + b2 x2 + b2 x3 + . . .)

and an mh root in the required form is only possible when n = vm, v being a positive whole number.

The infinite series R proceeding according to whole powers of x, whose mth root has been transformed into a similar infinite series, may also have all its coefficients after a certain term, equal to zero, so that it becomes nothing more than a whole function of a of the nth degree. According to what has preceded then, the_mth root of a whole function of x, may, with the above mentioned exception, be always transformed into a series proceeding according to whole powers of x.

We can now transform an irrational broken function into an infinite series proceeding according to whole powers of x, and that in all cases, while the coefficients are still perfectly general, although the transformation will admit of exception for particular values of the coefficients, i. e. when certain coefficients are equal to zero. Namely, when R and S are whole functions R of x, and therefore a broken rational function of x, we have S /(R:S)="/R: "/S, where the dividend and divisor on the right, and therefore also the quotient itself can be transformed into a similar series, at least as long as the coefficients are quite general and none of them has as yet received the value zero. The same holds for other compound expressions, e. g. for /R/S, &c. &c.

SECTION 47.

We can consequently always transform any expression anywise compounded of whole functions of x, or infinite series proceeding according to whole powers of x, as far as we yet know of the existence of any such expression (that is, provided no general power y and no general logarithm logy occurs, whose signification in calculation we have not yet been able to establish in these pages,) as long as the coefficients of the given infinite series or whole functions of x, are still perfectly general and have not yet received any particular value in cyphers, we can always transform such an expression into a series proceeding according to whole powers of x, which is " equal" to it in the

which

* For the expressions for A,, or A1, or A2, &c. take the form form always shews that the particular case which leads to it is not contained in the general calculation, and that consequently the calculation must be conducted in a particular manner for this particular case. No one can in such a case

seriously believe that the coefficient is infinitely great.

sense of (sect. 3), i. e. which may be unconditionally substituted for it in all "calculations" (sect. 6), without our having to fear contradictions on the part of the laws of operation.

There are therefore finite expressions which can be transformed, or, to employ the usual phrase, developed into general infinite series. From this it follows conversely that there will be some general infinite series which are "equal" (sect. 3) to a finite expression, i. e. for which a finite expression can be substituted in all "calculations" without our having to fear any contradictions on the part of the laws of operation.

This finite expression which can be substituted for such a general series, is commonly called the sum of the infinite series; and to find it, is usually termed "to sum the series." Such a sum only exists in very rare cases, but we often transform an infinite series into an expression compounded out of other infinite series, and we say that the first series is summed, when the latter series admit of more convenient treatment, and have been represented by simple symbols (as a, e, sin a, cos x, &c.)

We can consequently only speak of the summation of an infinite series, when it is still conceived as perfectly general, i. e. when it proceeds according to the powers of any expression x which is still general, i. e. which is still a mere supporter of the symbols of operation, and by which we do not as yet suppose any determinate value in cyphers to be represented*.

SECTION 48.

If in an infinite series proceeding according to whole powers of x, a determinate value in cyphers be given to x, we obtain an expression consisting of an infinite number of terms (in which the terms are still considered as perfectly general, or have already received certain values in cyphers), which has no longer the form of whole functions of x, and with which we are therefore no longer able to "calculate" according to the laws of the whole functions of a, but which has still the form of an algebraical sum of an infinite number of terms (whose terms must also follow in infinitum a determinately enunciated law, since it cannot otherwise be considered as given in infinitum,) so that we can still apply the laws of algebraical sums for the purpose of calculating with them.

But if such series have to be multiplied or divided by one another, we have usually no law for arranging the terms of the result, i. e. we have no determinately prescribed form to which the result is to be reduced; and it is consequently a very precarious affair to "calculate" with them (according to sect. 6); and if a calculation with them succeeds, i. e. leads to a desired

*We shall soon prove the existence of numerical infinite series, and then shew that although they have no sum as has been defined in the text, yet they have a value, and that though we cannot sum, we can evaluate them.