SERIES
, in Algebra, denotes a rank or progression of quantities or terms, which usually proceed according to some certain law. As the Series 1 + (1/2) + (1/4) + (1/8) + (1/16) &c, or the Series, 1 + (1/2) + (1/3) + (1/4) + (1/5) &c. where the former is a geometrical Series, proceeding by the constant division by 2, or the denominators multiplied by 2; and the latter is an harmonical Series, being the reciprocals of the arithmetical Series 1, 2, 3, 4, &c, or the denominators being continually increased by 1.
The doctrine and use of Series, one of the greatest improvements of the present age, we owe to Nicholas Mercator; though it seems he took the first hint of it from Dr. Wallis's Arithmetic of Infinites; but the genius of Newton first gave it a body and a form.
It is chiefly useful in the quadrature of curves; where, as we often meet with quantities which cannot be expressed by any precise definite numbers, such as is the ratio of the diameter of a circle to the circumference, we are glad to express them by a Series, which, infinitely continued, is the value of the quantity sought, and which is called an Infinite Series.
Infinite Series commonly arise, either from a continued division, as was practised by Mercator, or the extraction of roots, as first performed by Newton, who also explained other general ways for the expanding of quantities into infinite Series, as by the binomial theorem. Thus, to divide 1 by 3, or to expand the fraction 1/3 into an infinite Series; by division in decimals in the ordinary way, the series is 0.3333 &c, or (3/10) + (3/100) + (3/1000) + (3/10000) &c, where the law of continuation is manifest. Or, if the same fraction <*>/3 be set in this form 1/(2 + 1), and division be performed in the algebraic manner, the quotient will be &c. Or, if it be expressed in this form 1/3 = 1/(4 - 1), by a like division there will arise the Series, &c. And, thus, by dividing 1 by 5 - 2, or 6 - 3, or 7 - 4, &c, the Series answering to the fraction 1/3, may be found in an endless variety of infinite Series; and the finite quantity 1/3 is called the value or radix of the Series, or also its sum, being the number or sum to which the Series would amount, or the limit to which it would tend or approximate, by summing up its terms, or by collecting them together one after another.
In like manner, by dividing 1 by the algebraic sum a + c, or by a - c, the quotient will be in these two cases, as below, viz, &c. where the terms of each Series are the same, and they differ only in this, that the signs are alternately positive and negative in the former, but all positive in the latter.
And hence, by expounding a and c by any numbers whatever, we obtain an endless variety of infinite Series, whose sums or values are known. So, by taking a or c equal to 1 or 2 or 3 or 4, &c, we obtain these Series, and their values; &c.
And hence it appears, that the same quantity or radix may be expressed by a great variety of infinite Series, or that many different Series may have the same radix or sum.
Another way in which an infinite Series arises, is by the extraction of roots. Thus, by extracting the square root of the number 3 in the common way, we obtain its value in a series as follows, viz, &c; in which way of resolution the law of the progression | of the Series is not visible, as it is when sound by division. And the square root of the algebraic quantity a^{2} + c^{2} gives &c.
And a 3d way is by Newton's binomial theorem, which is a universal method, that serves for all sorts of quantities, whether fractional or radical ones: and by this means the same root of the last given quantity becomes &c. where the law of continuation is visible.
See Extraction of Roots, and Binomial Theorem.
From the specimens above given, it appears that the signs of the terms may be either all plus, or alternately plus and minus. Though they may be varied in many other ways. It also appears that the terms may be either continually smaller and smaller, or larger and larger, or else all equal. In the first case therefore the Series is said to be a decreasing one, in the 2d case an increasing one, and in the 3d case an equal one. Also the first Series is called a converging one, because that by collecting its terms successively, taking in always one term more, the successive sums approximate or converge to the value or sum of the whole infinite Series. So, in the Series , &c, the first term 1/3 is too little, or below 1/2 which is the value or sum of the whole infinite Series proposed; the sum of the first two terms (1/3) + (1/9) is 4/9 = .4444 &c, is also too little, but nearer to 1/2 or .5 than the former; and the sum of three terms (1/3) + (1/9) + (1/27) is 13/27 = .481481 &c, is nearer than the last, but still too little; and the sum of four terms &c. which is again nearer than the former, but still too little; which is always the case when the terms are all positive. But when the converging Series has its terms alternately positive and negative, then the successive sums are alternately too great and too little, though still approaching nearer and nearer to the final sum or value. Thus in the Series &c, are too great, four terms &c, are too great, and so on, alternately too great and too small, but every succeeding sum still nearer than the former, or converging.
In the second case, or when the terms grow larger and larger, the Series is called a diverging one, because that by collecting the terms continually, the successive sums diverge, or go always farther and farther from the true value or radix of the Series; being all too great when the terms are all positive, but alternately too great and too little when they are alternately positive and negative. Thus, in the Series &c. the first term + 1 is too great, two terms 1 - 2 = - 1 are too little, three terms 1 - 2 + 4 = + 3 are too great, four terms 1 - 2 + 4 - 8 = - 5 are too little, and so on continually, after the 2d term, diverging more and more from the true value or radix 1/3, but alternately too great and too little, or positive and negative. But the alternate sums would be always more and more too great if the terms were all positive, and always too little if negative.
But in the third case, or when the terms are all equal, the Series of equals, with alternate signs, is called a neutral one, because the successive sums, found by a continual collection of the terms, are always at the same distance from the true value or radix, but alternately positive and negative, or too great and too little. Thus, in the Series &c, the first term 1 is too great, two terms 1 - 1 = 0 are too little, three terms 1 - 1 + 1 = 1 too great, four terms 1 - 1 + 1 - 1 = 0 too little, and so on continually, the successive sums being alternately 1 and 0, which are equally different from the true value or radix 1/2, the one as much above it, as the other below it.
A Series may be terminated and rendered finite, and accurately equal to the sum or value, by assuming the supplement, after any particular term, and combining it with the foregoing terms. So, in the Series (1/2) - (1/4) + (1/8) - (1/16) &c, which is equal to 1/3, and found by dividing 1 by 2 + 1, after the first term, 1/2, of the quotient, the remainder is - (1/2), which divided by 2 + 1, or 3, gives - (1/6) for the supplement, which | combined with the first term 1/2, gives (1/2) - (1/6) = 1/3 the true sum of the Series. Again, after the first two terms (1/2) - (1/4), the remainder is + (1/4), which divided by the same divisor 3, gives 1/12 for the supplement, and this combined with those two terms (1/2) - (1/4), makes or 1/3 the same sum or value as before. And in general, by dividing 1 by a + c, there is obtained ; where, stopping the division at any term as c^{n}/a^{n + 1}, the remainder after this term is c^{n + 1}/a^{n + 1}, which being divided by the same divisor a + c, gives c^{n + 1}/(a^{n + 1} (a + c)) for the supplement as above.
The Law of Continuation.—A Series being proposed, one of the chief questions concerning it, is to find the law of its continuation. Indeed, no universal rule can be given for this; but it often happens that the terms of the Series, taken two and two, or three and three, or in greater numbers, have an obvious and simple relation, by which the Series may be determined and produced indefinitely. Thus, if 1 be divided by 1 - x, the quotient will be a geometrical progression, viz, 1 + x + x^{2} + x^{3} &c, where the succeeding terms are produced by the continual multiplication by x. In like manner, in other cases of division, other progressions are produced.
But in most cases the relation of the terms of a Series is not constant, as it is in those that arise by division. Yet their relation often varies according to a certain law, which is sometimes obvious on inspection, and sometimes it is found by dividing the successive terms one by another, &c. Thus, in the Series 1 + ((2/3)x) + ((8/15)x^{2}) + ((16/35)x^{3}) + ((128/315)x^{4})) &c, by dividing the 2d term by the 1st, the 3d by the 2d, the 4th by the 3d, and so on, the quotients will be (2/3)x, (4/5)x, (6/7)x, (8/9)x, &c; and therefore the terms may be continued indefinitely by the successive multiplication by these fractions. Also in the following Series 1 + ((1/6)x)) + ((3/40)x^{2}) + ((5/128)x^{3}) + ((35/1152)x^{4}) &c, by dividing the adjacent terms successively by each other, the Series of quotients is (1/6)x, (9/20)x, (25/42)x, (49/72)x, &c, or (1.1/2.3)x, (3.3/4.5)x, (5.5/6.7)x, (7.7/8.9)x, &c; and therefore the terms of the Series may be continued by the multiplication of these fractions.
Another method of expressing the law of a Series, is one that defines the Series itself, by its general term, shewing the relation of the terms generally by their distances from the beginning, or by differential equations. To do this, Mr. Stirling conceives the terms of the Series to be placed as so many ordinates on a right line given by position, taking unity as the common interval between these ordinates. The terms of the Series he denotes by the initial letters of the alphabet, A, B, C, D, &c; A being the first, B the 2d, C the 3d, &c: and he denotes any term in general by the letter T, and the rest following it in order by T′, T″, T‴, T′′′′, &c; also the distance of the term T from any given term, or from any given intermediate point between two terms, he denotes by the indeterminate quantity z: so that the distances of the terms T′, T″, T‴, &c, from the said term or point, will be z + 1, z + 2, z + 3, &c; because the increment of the absciss is the common interval of the ordinates, or terms of the Series, applied to the absciss.
These things being premised, let this Series be proposed, viz, 1, (1/2)x, (3/8)x^{2}, (5/16)x^{3}, (35/128)x^{4}, (63/256)x^{5}, &c; in which it is found, by dividing the terms by each other, that the relations of the terms are, , &c: then the relation in general will be defined by the equation , where z denotes the distance of T from the first term of the SeriesFor by substituting 0, 1, 2, 3, 4, &c, successively instead of z, the same relations will arise as in the proposed Series above. In like manner, if z be the distance of T from the 2d term of the Series, the equation will be , as will appear by substituting the numbers - 1, 0, 1, 2, 3, &c, successively for z. Or, if z denote the place or number of the term T in the Series, its successive values will be 1, 2, 3, 4, &c, and the equation or general term will be .
It appears therefore, that innumerable differential equations may define one and the same Series, according to the different points from whence the origin of the absciss z is taken. And, on the contrary, the same equation defines innumerable different Series, by taking different successive values of z. For in the equation , which defines the foregoing Series | when 1, 2, 3, 4, &c are the successive values of the abscisses; if 1 1/2, 2 1/2, 3 1/2, 4 1/2, &c, be successively substituted for z, the relations of the terms arising will be, , &c, from whence will arise the Series A, (2/3)Ax, (8/15)Ax^{2}, (16/35)Ax^{3}, (128/315)Ax^{4}, &c, which is different from the former.
And thus the equation will always determine the Series from the given values of the absciss and of the first term, when the equation includes but two terms of the Series, as in the last example, where the first term being given, all the rest will be given.
But when the equation includes three terms, then two must be given; and three must be given, when it includes four; and so on. So, if there be proposed the Series x, (1/6)x^{3}, (3/40)x^{5}, (5/128)x^{7}, (35/1152)x^{9}, &c, where the relations of the terms are, , &c. the equation defining this Series will be , where the successive values of z are 1, 2, 3, 4, &c. See Stirling's Methodus Differentialis, in the introduction.
This may suffice to give a notion of these differential equations, defining the nature of Series. But as to the application of these equations in interpolations, and finding the sums of Series, it would require a treatise to explain it. We must therefore refer to that excellent one just quoted, as also to De Moivre's Miscellanea Analytica; and several curious papers by Euler in the Acta Petropolitana.
A Series often converges so slowly, as to be of no use in practice. Thus, if it were required to find the sum of the Series (1/(1.2)) + (1/(3.4)) + (1/(5.6)) + (1/(7.8)) + (1/(9.10)) &c, which Lord Brouncker found for the quadrature of the hyperbola, true to 9 figures, by the mere addition of the terms of the Series; Mr. Stirling computes that it would be necessary to add a thousand millions of terms for that purpose; for which the life of man would be too short. But by that gentleman's method, the sum of the Series may be found by a very moderate computation. See Method. Differ. pa. 26.
Series are of various kinds or descriptions. So,
An Ascending Series, is one in which the powers of the indeterminate quantity increase; as 1 + ax + bx^{2} + cx^{3} &c. And a
Descending Series, is one in which the powers decrease, or else increase in the denominators, which is the same thing; as 1 + ax^{- 1} + bx^{- 2} + cx^{- 3} &c, or 1 + (a/x) + (b/x^{2}) + (c/x^{3}) &c.
A Circular Series, which denotes a Series whose sum depends on the quadrature of the circle. Such is the Series 1 + (1/3) + (1/5) - (1/7) + (1/9) &c: See Demoivre Miscel. Analyt. pa. 111, or my Mensur. pa. 119. Or the sum of the Series 1 + (1/4) + (1/9) + (1/16) + (1/25) &c, continued ad insinitum, according to Euler's discovery.
Continued Fraction or Series, is a fraction of this kind, to infinity, . The first Series of this kind was given by Lord Brouncker, first president of the Royal Society, for the quadrature of the circle, as related by Dr. Wallis, in his Algebra, pa. 317. His series is , which denotes the ratio of the square of the diameter of a circle to its area. Mr. Euler has treated on this kind of Series, in the Petersburgh Commentaries, vol. 11, and in his Analys. Infinit. vol. 1, pa. 295, where he shews various uses of it, and how to transform ordinary fractions and common Series into continued fractions. A common fraction is transformed into a continued one, after the manner of seeking the greatest common measure of the numerator and denominator, by dividing the greater by the less, and the last divisor always by the last remainder. Thus to change 1461/59 to a continued fraction. |
Converging Series, is a Series whose terms continually decrease, or the successive sums of whose terms approximate or converge always nearer to the ultimate sum of the whole Series. And, on the contrary, a
Diverging Series, is one whose terms continually increase, or that has the successive sums of its terms diverging, or going off always the farther, from the sum or value of the Series.
Determinate Series, is a Series whose terms proceed by the powers of a determinate quantity; as 1 + (1/2) + (1/2^{2}) + (1/2^{3}) + &c. If that determinate quantity be unity, the Series is said to be determined by unity. De Moivre, Miscel. Analyt. pa. 111. And an
Indeterminate Series is one whose terms proceed by the powers of an indeterminate quantity x; as x + ((1/2)x^{2}) + ((1/3)x^{3}) + ((1/4)x^{4}) &c; or sometimes also with indeterminate exponents, or indeterminate coefficients.
The Form of a Series, is used for that affection of an indeterminate Series, such as ax^{n} + bx^{n + r} + cx^{n + 2r} + dx^{n + 3r} &c, which arises from the different values of the indices of x. Thus, If n = 1, and r = 1, the Series will take the form ax + bx^{2} + cx^{3} + dx^{4} &c. If n = 1, and r = 2, the form will be ax + bx^{3} + cx^{5} + dx^{7} &c. If n = 1/2, and r = 1, the form is ax^{1/2} + bx^{3/2} + cx^{3/2} + dx^{7/2} &c. And If n = 0, and r = - 1, the form will be a + bx^{- 1} + cx^{-2} + dx^{- 3} &c.
When the value of a quantity cannot be found exactly, it is of use in algebra, as well as in common arithmetic, to seek an approximate value of that quantity, which may be useful in practice. Thus, in arithmetic, as the true value of the square root of 2 cannot be assigned, a decimal fraction is found to a sufficient degree of exactness in any particular case; which decimal fraction is in reality, no more than an infinite series of fractions converging or approximating to the true value of the root sought. For the expression √2 = 1.414213 &c, is equivalent to this &c; or supposing x = 10, to this &c, which last Series is a particular case of the more general indeterminate Series ax^{n} + bx^{n + r} + cx^{n + 2r} &c, viz, when n = 0, r = - 1, and the coefficients a = 1, b = 4, c = 1, d = 4, &c.
But the application of the notion of approximations in numbers, to species, or to algebra, is not so obvious. Newton, with his usual sagacity, took the hint, and prosecuted it; by which were discovered general methods in the doctrine of infinite Series, which had before been treated only in a particular manner, though with great acuteness, by Wallis and a few others. See Newton's Method of Fluxions and Infinite Series, with Colson's Comment; as also the Analysis per Æquationes Numero Terminorum Infinitas, published by Jones in 1711, and since translated and explained by Stewart, together with Newton's Tract on Quadratures, in 1745. To these may be added Maclaurin's Algebra, part 2, chap. 10, pa. 244; and Cramer's Analyse des Lignes Courbes Algebraiques, chap. 7, pa. 148; and many other authors.
Among the various methods for determining the value of a quantity by a converging Series, that seems preferable to the rest, which consists in assuming an indeterminate Series as equal to the quantity whose value is sought, and afterwards determining the values of the terms of this assumed Series. For instance, suppose a logarithm were given, to find the natural number answering to it. Suppose the logarithm to be z, and the corresponding number sought 1 + x: then by the nature of logarithms and fluxions, . Now assume a Series for the value of the unknown quantity x, and substitute it and its fluxion instead of x and x^{.} in the last equation, then determine the assumed coefficients by comparing or equating the like terms of the equation. Thus, &c; hence, comparing the like terms of these two values of x^{.}, there arises a = 1, b = 1/2, c = 1/6, d = 1/24, &c; which values being substituted for a, b, c, &c, in the assumed Series ax + bx^{2} + cx^{3} &c, it gives &c; and consequently the number sought will be &c.
But the indeterminate Series az + bz^{2} + cz^{3} &c, was here assumed arbitrarily, with regard to its exponents 1, 2, 3, &c, and will not succeed in all cases, because some quantities require other forms for the exponents. For instance, if from an arc given, it were required to find the tangent. Let x = the tangent, and z the arc, the radius being = 1. Then, from the nature of the circle we shall have . Now if, to find the value of x, we suppose &c, and proceed as before, we shall find all the alternate coefficients b, d, f, &c, or those of the even powers of z, to be each = 0; and therefore the Series assumed is not of a proper form. | But supposing , &c, then we find a = 1, b = 1/3, c = 2/15, d = 17/315, &c, and consequently &c. And other quantities require other forms of Series.
Now to find a proper indeterminate Series in all cases, tentatively, would often be very laborious, and even impracticable. Mathematicians have therefore endeavoured to find out a general rule for this purpose; though till lately the method has been but imperfectly understood and delivered. Most authors indeed have explained the manner of finding the coefficients a, b, c, d, &c, of the indeterminate Series ax^{n} + bx^{n + r} + cx^{n + 2r} &c, which is easy enough; but the values of n and r, in which the chief difficulty lies, have been assigned by many in a manner as if they were self-evident, or at least discoverable by an easy trial or two, as in the last example.
As to the number n, Newton himself has shewn the method of determining it, by his rule for finding the first term of a converging Series, by the application of his parallelogram and ruler. For the particulars of this method, see the authors above cited; see also PARALLELOGRAM.
Taylor, in his Methodus Incrementorum, investigates the number r; but Stirling observes that his rule sometimes fails. Lineæ Tert. Ordin. Newton. pa. 28. Mr. Stirling gives a correction of Taylor's rule, but says he cannot affirm it to be universal, having only found it by chance. And again
Gravesande observes, that though he thinks Stirling's rule never leads into an error, yet that it is not perfect. See Gravesande, De Determin. Form. Seriei Infinit. printed at the end of his Matheseos Universalis Elementa. This learned professor has endeavoured to rectify the rule. But Cramer has shewn that it is still defective in several respects; and he himself, to avoid the inconveniences to which the methods of former authors are subject, has had recourse to the first principles of the method of infinite Series, and has entered into a more exact and instructive detail of the whole method, than is to be met with elsewhere; for which reason, and many others, his treatise deserves to be particularly recommended to beginners.
But it is to be observed, that in determining the value of a quantity by a converging Series, it is not always necessary to have recourse to an indeterminate Series: for it is often better to find it by division, or by extraction of roots. See Newton's Meth. of Flux. and Inf. Series, above cited. Thus, if it were required to find the arc of a circle from its given tangent, that is, to find the value of z in the given fluxional equation, , by an infinite Series: dividing x^{.} by 1 + xx, the quotient will be the Series ; and taking the fluents of the terms, there results &c, which is the Series often used for the quadrature of the circle. If x = 1, or the tangent of 45°, then will &c = the length of an arc of 45°, or 1/8 of the circumference, to the radius 1, or 1/4 of the circumference to the diameter 1. Consequently, if 1 be the diameter, then 1 - (1/3) + (1/5) - (1/7) &c will be the area of the circle, because 1/4 of the circumference multiplied by the diameter, gives the area of the circle. And this Series was first given by Leibnitz and James Gregory.
See the form of the Series for the binomial theorem, determined, both as to the coefficients and exponents, in my Tracts, vol. 1, pa. 79.
Harmonical Series, the reciprocal of arithmeticals. See Harmonical.
Hyperbolic Series, is used for a Series whose sum depends upon the quadrature of the hyperbola. Such is the Series (1/1) + (1/2) + (1/3) + (1/4) &c. De Moivre's Miscel. Analyt. pa. 111.
Interpolation of Series, the inserting of some terms between others, &c. See Interpolation.
Interscendent Series. See Interscendent.
Mixt Series, one whose sum depends partly on the quadrature of the circle, and partly on hit of the hyperbola. De Moivre, Miscel. Analyt. pa. 111.
Recurring Series, is used for a Series which is so constituted, that having taken at pleasure any number of its terms, each following term shall be related to the same number of preceding terms according to a constant law of relation. Thus, in the following Series,
a | b | c | d | e | f | |
1 + | 2x + | 3x^{2} + | 10x^{3} + | 34x^{4} + | 97x^{5} | &c, |
The quantities 3x - 2x^{2} + 5x^{3}, taken together and connected by their proper signs, form what De Moivre calls the index, or the scale of relation; though sometimes the bare coefficients 3 - 2 + 5 are called the scale of relation. And the scale of relation subtracted from unity, is called the differential scale. On the subject of Recurring Series, see De Moivre's Miscel. Analyt. pa. 27 and 72, and his Doctrine of Chances, 3d edit. pa. 220; also Euler's Analys. Infinit. tom. 1, pa. 175.
Having given a recurring Series, with its scale of relation, the sum of the whole infinite Series will also be given. For instance, suppose a Series | a + bx + cx^{2} + dx^{3} &c, where the relation between the coefficient of any term and the coefficients of any two preceding terms may be expressed by f - g; that is, , &c; then will the sum of the Series, infinitely continued, be (a + (b - fa) x)/(1 - fx + gx^{2}).
Thus, for example, assume 2 and 5 for the coefficients of the first two terms of a recurring Series; and suppose f and g to be respectively 2 and 1; then the recurring Series will be 2 + 5x + 8x^{2} + 11x^{3} + 14x^{4} + 18x^{5} &c, and its sum . For the proof of which divide 2 + x by (1 - x)^{2}, and there arises the said Series 2 + 5x + 8x^{2} + 11x^{3} &c. And similar rules might be derived for more complex cases.
De Moivre's general rule is this: 1. Take as many terms of the Series as there are parts in the scale of relation. 2. Subtract the scale of relation from unity, and the Remainder is the differential scale. 3. Multiply the terms taken in the Series by the differential scale, beginning at unity, and so proceeding orderly, remembering to leave out what would naturally be extended beyond the last of the terms taken. Then will the product be the numerator, and the differential scale will be the denominator of the fraction expressing the sum required.
But it must here be observed, that when the sum of a recurring Series extended to infinity, is found by De Moivre's rule, it ought to be supposed that the Series converges indefinitely, that is, that the terms may become less than any assigned quantity. For if the Series diverge, that is, if its terms continually increase, the rule does not give the true sum. For the sum in such case is infinite, or greater than any given quantity, whereas the sum exhibited by the rule, will often be finite. The rule therefore in this case only gives a fraction expressing the radix of the Series, by the expansion of which the Series is produced. Thus 1/((1 - x)^{2}) by expansion becomes the recurring Series 1 + 2x + 3x^{2} &c, whose scale of relation is 2 - 1, and its sum by the rule will be , the quantity from which the Series arose. But this quantity cannot in all cases be deemed equal to the infinite Series 1 + 2x + 3x^{2} &c: for stop where you will, there will always want a supplement to make the product of the quotient by the divisor equal to the dividend. Indeed when the Series converges infinitely, the supplement, diminishing continually, becomes less than any assigned quantity, or equal to nothing; but in a diverging Series, this supplement becomes insinitely great, and the Series deviates indefinitely from the truth. See Colson's Comment on Newton's Method of Fluxions and Infinite Series, pa. 152; Stirling's Method. Differ. pa. 36; Bernoulli de Serieb. Insin. pa. 249; and Cramer's Analyse des Lignes Courbes, pa. 174.
A recurring Series being given, the sum of any finite number of the terms of that Series may be found. This is prob. 3, pa. 73, De Moivre's Miscel. Analyt. and prob. 5, pa. 223 of his Doctrine of Chances. The solution is effected, by taking the difference between the sums of two infinite Series, differing by the terms answering to the given number; viz, from the sum of the whole infinite Series, commencing from the beginning, subtract the sum of another insinite number of terms of the same Series, commencing after so many of the first terms whose sum is required; and the difference will evidently be the sum of that number of terms of the Series. For example, to find the sum of n terms of the infinite geometrical Series a + ax + ax^{2} + ax^{3} &c. Here are two insinite Series; the one beginning with a, and the other with ax^{n}, which is the next term after the first n terms of the original Series. By the rule, the sum of the first infinite progression will be a/(1 - x), and the sum of the second ax^{n}/(1 - x); the difference of which is (a - ax^{n})/(1 - x), which is therefore the sum of the first n terms of the Series. This quantity (a - ax^{n})/(1 - x) is equal to (ax^{n} - a)/(x - 1) which last expression, putting , will be equivalent to this, (lx - a)/(x - 1), which is the common rule for finding the sum of any geometric progression, having given the first term a, the last term l, and the ratio x. See Miscel. Analyt. pa. 167, 168.
In a recurring Series, any term may be obtained whose place is assigned. For after having taken so many terms of the Series as there are terms in the scale of relation, the Series may be protracted till it reach the place assigned. But when that place is very distant from the beginning of the Series, the continuing the terms is very laborious; and therefore other methods have been contrived. See Miscel. Analyt. pa. 33; and Doctrine of Chances, pa. 224.
These questions have been resolved in many cases, besides those of recurring Series. But as there is no universal method for the quadrature of curves, neither is there one for the summation of Series; indeed there is a great analogy between these things, and similar difficulties arising in both. See the authors above cited.
The investigation of Daniel Bernoulli's method for finding the roots of algebraic equations, which is inserted in the Petersburgh Acts, tom. 3, pa. 92, depends upon the doctrine of recurring Series. See Euler's Analysis Infinitorum, tom 1, pa. 276.
Reversion of Series. See Reversion of Series.
Summable Series, is one whose sum can be accurately found. Such is the Series 1/2 + 1/4 + 1/8 &c, the sum of which is said to be unity, or, to speak more accurately, the limit of its sum is unity or 1.
An indefinite number of summable insinite Series | may be assigned: such are, for instance, all infinite recurring converging Series, and many others, for which, consult De Moivre, Bernoulli, Stirling, Euler, and Maclaurin; viz, Miscel. Analyt. pa. 110; De Serieb. Infinit. passim; Method. Different. pa. 34; Acta Petrop passim; Fluxions, art. 350.
The obtaining the sums of infinite Serieses of fractions has been one of the principal objects of the modern method of computation; and these sums may often be found, and often not. Thus the sums of the two following Series of geometrical progressionals are easily found to be 1 and 1/2, viz, &c.
But the Serieses of fractions that occur in the solution of problems, can seldom be reduced to geometric progressions; nor can any general rule, in cases so infinitely various, be given. The art here, as in most other cases, is only to be acquired by examples, and by a careful observation of the arts used by great authors in the investigation of such Series of fractions as they have considered. And the general methods of infinite Series, which have been carried so far by De Moivre, Stirling, Enler, &c, are often found necessary to determine the sum of a very simple Series of fractions. See the quotations above.
The sum of a Series of fractions, though decreasing continually, is not always sinite. This is the case of the Series 1/1 + 1/2 + 1/3 + 1/4 + 1/5 &c, which is the harmonic Series, consisting of the reciprocals of arithmeticals, the sum of which exceeds any given number whatever; and this is shewn from the analogy between this progression and the space comprehended by the common hyperbola and its asymptote; though the same may be shewn also from the nature of progressions. See James Bernoulli, de Seriebus Infin. But, what is curious, the square of it is finite, for if the same terms of the harmonic Series, 1/1 + 1/2 + 1/3 &c, be squared, forming the Series 1/1 + 1/4 + 1/9 &c, being the reciprocals of the squares of the natural Series of numbers; the sum of this Series of fractions will not only be limited, but it is remarkable that this sum will be precisely equal to the 6th part of the number which expresses the ratio of the square of the circumference of a circle to the square of its diameter. That is, if c denote 3.14159 &c, the ratio of the circumference to the diameter, then is &c. This property was first discovered by Euler; and his investigation may be seen in the Acta Petrop. vol. 7. And Maclaurin has since observed, that this may easily be deduced from his Fluxions, art. 822. Philos. Trans. numb. 469.
It would require a whole treatise to enumerate the various kinds of Series of fractions which may or may not be summed. Sometimes the sum cannot be assigned, either because it is infinite, as in the harmonic Series 1/1 + 1/2 + 1/3 + 1/4 &c, or although its sum be finite (as in the Series 1/1 + 1/4 + 1/9 &c), yet its sum cannot be assigned in finite terms, or by the quadrature of the circle or hyperbola, which was the case of this Series before Euler's discovery; but yet the sum of any given number of the terms of the Series may be expeditiously found, and the whole sum may be assigned by approximation, independent of the circle. See Stirling's Method. Different. and De Moivre's Miscel. Analyt.
Besides the Serieses of fractions, the sums of which converge to a certain quantity, there sometimes occur others, which converge by a continued multiplication. Of this kind is the Series found by Wallis, for the quadrature of the circle, which he expresses thus, , where the character □ denotes the ratio of the square of the diameter to the area of the circle. Hence the denominator of this fraction, is to its numerator, both insinitely continued, as the circle is to the square of the diameter. It may farther be observed that this Series is equivalent to (9/8) X (25/24) X (49/48) X &c, or to (3^{2}/(3^{2} - 1)) X (5^{2}/(5^{2} - 1)) X (7^{2}/(7^{2} - 1)) X &c, that is, the product of the squares of all the odd numbers 3, 5, 7, 9, &c, is to the product of the same squares severally diminished by unity, as the square of the diameter is to the area of the circle. See Arithmet. Insinit. prop. 191, Oper. vol. 1, pa. 469. Id. Oper. vol. 2, pa. 819. And these products of fractions, and the like quantities arising from the continued multiplication of certain factors, have been particularly considered by Euler, in his Analysis Infinit. vol. 1, chap. 15, pa. 221.
For an easy and general method of summing all alternate Series, such as a - b + c - d &c, see my Tracts, vol. 1, pa. 11; and in the same vol. may be seen many other curious tracts on infinite Series.
Summation of Insinite Series, is the finding the value of them, or the radix from which they may be raised. For which, consult all the authors upon this science.
To find an infinite Series by extracting of roots; and to find an infinite Series by a presupposed Series; see Quadrature of the Circle.
To extract the roots of an infinite Series, see EXTRACTION of Roots.
To raise an infinite Series to any power, see INVOLUTION, and Power.
Transcendental Series. See Transcendental.
There are many other important writings upon the subject of Infinite Series, besides those above quoted. A very good elementary tract on this science is that of James Bernoulli, intituled, Tractatus de Seriebus Infini- | tis, and annexed to his Ars Conjectandi, published in 4to, 1713.