CHAPTER II. OF INFINITE SERIES. (Art. 136.) An infinite series is a continued rank, or progression of quantities in regular order, in respect both to magnitudes and signs, and they usually arise from the division of one quantity by another. The roots of imperfect powers, as shown by the examples in the last article, produce one class of infinite series. Some of the examples under (Art. 121.) show the geometrical infinite series. Examples in common division may produce infinite series for quotients; or, in other words, we may say the division is continuous. Thus, 10 divided by 3, and carried out in decimals, give 3.3333, &c. without end, and the sum of such a series is 31. (Art. 121.) (Art. 137.) Two series may appear very different which arise from the same source; thus 1, divided by 1+a, gives, as we may see by actual division, as follows: These two quotients appear very different, and in respect to single terms are so; but in the division there is always a remainder, and either quotient is incomplete without the remainder for a numerator and the divisor for a denominator, and when these are taken into consideration the two quotients will be equal. We may clearly illustrate this by the following example: Divide 3 by 1+2, the quotient is manifestly 1; but suppose them literal quantities, and the division would appear thus: 1+2)3 (3-6+12, &c. -6 -6-12 12 12+24 -24 Again, divide the same, having the 2 stand first. 2+1)3(+, &c. Now let us take either quotient, with the real value of its remainder, and we shall have the same result. Thus, 3+12=15; and -6, and the remainder -24 divided by 3 gives -8, which makes -14; hence, the whole quotient is 1. Again, Hence, and - ==1, the proper quotient. If we more closely examine the terms of these quotients, we shall discover that one is diverging, the other converging, and by the same ratio 2, and in general this is all a series can show, the degree of convergency. (Art. 138.) We convert quantities into series by extracting the roots of imperfect powers, as by the binomial, and by actual division, thus: Thus, a+x)a (1-a+a2-ast, &c. Observe that these two examples are the same, except the signs to x; when that sign is plus the signs in the series will be alternately plus and minus; when minus, all will be plus. 3. What series will arise from 1+? 1-x Ans. 1+2x+2x2+2x2, &c. Observe that in this case the series commences with 2x. The unit is a proper quotient, and the series arises alone 2x from the remainder after the quotient 1 is obtained. Observe, in this example, the term x, in the numerator, does not find a place in the operation; it will be always in 2 5. What series will arise from dividing 1 by 1-a+a2, Ans. 1+a-a3-a2+a+a-a-a10, &c. In this example, observe that the signs are not alternately plus and minus, but two terms in succession plus, then two minus; this arises from there being two terms in place of one after the minus sign in the divisor. 6. What series will arise from a ? 1-r Ans. a+ar+ar2+ar3+ar, &c. Observe that this is the regular geometrical series, as appears in (Art, 118.) (Art. 139.) A fraction of a complex nature, or having 1-x compound terms, such as 1-2-32, may give rise to an infinite series, but there will be no obvious ratio between the terms. Some general relation, however, will exist between any one term and several preceding terms, which is called the scale of relation, and such a series is called a recurring series. Thus the preceding fraction, by actual division, gives 1+x+5x2+13x2+41x4+121x5, &c., a recurring series, which, when carried to infinity, will be equal to the fraction from which it is derived. CHAPTER III. SUMMATION OF SERIES. (Art. 140.) We have partially treated of this subject in geometrical progression, in (Art. 121.); the investigation is now more general and comprehensive, and the object in some respects different. There we required the actual sum of a given number of terms, or the sum of a converg. ing infinite series. Here the series may not be in the strictest sense geometrical, and we may not require the sum of the series, but what terms or fractional quantities will produce a series of a given convergency. The object then, is the converse of the last chapter; and for every geometrical series, our rule will be drawn from the sixth example in that chapter; that is, a 1-r a being the first term of any series, and r the ratio. We find the ratio by dividing any term by its preceding term. Hence, to find what fraction may have produced any geometrical series, we have the following rule: RULE. Divide the first term of the series by the algebraic difference between unity and the ratio. EXAMPLES. 1. What fraction will produce the series 2, 4, 8, 16, &c. Here a=2, and r=2; therefore, 2 Ans. 1-2 1 |