(5.) A series is a number of algebraical expressions, each of which is connected with those which precede it in some determinate manner. For example:-In the treatise on Elementary Algebra, we have had examples of series in the arithmetical and geometrical progressions. In the former case, each term is derived from the one preceding it by adding a certain known number called the common difference. In the latter case, each term is derived from the one preceding it by multiplying that term by a certain known number called the common ratio. Hence, a + (a + b) + (a +26) + (a + 36) + . . . are series. a+ar + ar2 + ar3 + . and 1+r+r2 + p3 +. Def.:-A series is called a finite series when it has an assignable last term. It is called an infinite series when, if we fix on any term whatever, there are terms beyond it. Thus, 1+r+ p2 + . +TM is a finite series. But 1+r + y2 + ... ... .... ad inf. is an infinite series, because if we take any term whatever-for instance, the 50th, or 500th, or 5000th-there are always terms beyond it. .... 6. To explain what is meant by a Convergent and a Divergent Series. Def.-If the sum of the terms of a series has an arithmetical limit when the number of terms is infinite, that series is convergent; if otherwise, it is divergent. If we divide 1 by 1 -r, we shall produce 1+ r+ r2 + . which series we can continue to produce to any number of terms whatever, Hence the fraction and the series 1+r+r2 + ad inf. are equivalent to each other; or 1 1-r .... ;=1+r+r2+r3 + ad infinitum. .... 1 1-r, Now, it has been already proved that if r 1, by taking a sufficiently large number of terms, the numerical value of the series can be made to approach to the numerical value of 1 1 1 -r to within any assignable limits. For instance, if r = then = 2; and if we take four term the series equals 1.875. If we take five terms it equals 1.9375; if six terms it equals 1.96875; and hence in the extreme case, when we suppose the number of terms to be infinitely large, the series is actually equal to 2. And hence if is less than 1, r =1+r+r2+ r3 + ad infinitum, 1 where by the sign we mean that the fraction is arithmetically equal to the 1 series. But if r is greater than 1, for instance, if equal to 2, the fraction equals -1; whereas if we take four terms, the series 15; if five terms, 31; if six terms, 63; and so on where there is no trace of approximation towards arithmetical equality between the series and the fraction. In the former case the series is said to be convergent, in the latter divergent; and if we include both cases in the expression, it must be understood that the sign: = signifies algebraically equivalent, not arithmetically equal. This explanation will be sufficient to enable the student to understand the meaning of the terms convergent and divergent, when applied to special series. The general questions that are suggested by series, and their convergency and divergency, belong to the higher parts of the science and many of them are still doubtful. 7. A test for ascertaining the Convergency of a given Series. We have already seen that— = 1 +r+ r2 + p3 +.... is convergent when r is 1. Hence, if we have A+B+C+D+.... and can show that BrA, C ≤ r2A, D Z r3A, &c. Then A+B+C+D+ ... < A(1 + r + r2 + p3 +. .). This latter is convergent if r is 1. And if so, the former must plainly be convergent too. This gives us a test for ascertaining whether a given series is convergent, which we shall find useful hereafter. The student must remember that, though all series which submit to this test are convergent, many may be convergent which do not submit to it. For instance, to ascertain whether the series <(and so on for the other terms; hence the given series 2 and this is convergent if ( 41, or if 8 / 2; and hence the given series is conver Hence, the above test of convergency holds good when the terms are alternately positive and negative. In the following pages we shall confine our attention to three series-the Binomial, the Exponential, and the Logarithmic. The reader will observe that the first term is a", that in each term the index of a diminishes by unity, while the power of b continually increases by unity, so that the sum of the indices of each term is n. Again, the coefficient of each term has for its denominator the continued products 1.2.3.... up to the index of b inclusive; and for numerator, the continued product n.(n − 1) (n − 2). . . . down to the index of a exclusive. Thus the coefficient of the term which involves an―rbr is 9. To prove the Binomial Theorem when n is a positive Integer. (a) To show that (a + b)" : = an+nan−16 +.... By actual multiplication, which is clearly of the same form as the assumption, i. e. this has m+1, wherever that has m. Hence, if the theorem is true for m, it must also be true for m+1. Now it is true for 4, .. it is true for 5, .. for 6, and so on; therefore it is always true for any positive whole number. N.B.-If a = :. (a+b) n = a" + nam¬+... 1, and b x, we of course have (a) (b) (1+ "=1+ n.(n 1) x2 + n(n-1) (n-2) (8). To show that— - 1.2.3 For, suppose (1 + x)" = 1 + nx + A2 x2 + A ̧ x3 + A1 x2 + . . . (c) It is plain, since (1 + x)" means (1+x) multiplied into itself n times, that this series is finite, so that we may employ the principle of indeterminate coefficients. In the series A2, A3, A4 . . . . . do not at all depend on x, and will therefore continue the same for all values of x, so that, for instance (1+ y)" =1+ny + Â„y2 + A ̧y3 † Â ̧ya + .... In equation (a), write x = y + z. Then (1 + y + z)n = 1 + n(y + z) + A2 (y + 2)2 + A ̧(y + ≈) 3 + . . . . which is true for all values of y and z. Again, if in equation (a) we take 1+y=a and ≈ = b, we have (1 + y + z)” = (1 + y)" + n(1 + y)n−1z + (E). Now, (d) and (e) are the same for all values of z, .. the coefficient of z in each must be the same, .. for all values of y; multiply both sides by I+y, .. n(1 + y)n =n + QA2y + 3A ̧y2 + 4A4Y3 + · But by equation, (c) n(1 + y)" = n + n2y + A2ny2 + Ã ̧ny3 + A1ny1 +.... .. n + n2y +nA2y2 + nA ̧y3 + nA4y4 + . =n+ (2A2 + n)y † (SA ̧ + 2A2)y2 + (4A4 + 3A ̧)y1 + . These expressions are true for all values of y. .... A1 = n(n − 1) (n − 2) ̧n(n − 1)(n − 2)(n − 3) 1.2.3.4 1.2 1.2.3 The student will observe the manner in which each successive coefficient is derived from the one that goes before it. He will easily see that if we look in the r— - Ith and pth terms, viz., A”¬¿1⁄2” − 1 + Arx” +. we should then have an equation— .... He will also observe that if n is a whole number when r is greater than n, there will be in the general term a factor n ·~+1+1=0; or x" is the last term of the n' (n − 1) -- 1.2 which is the Binomial Theorem when ʼn is a positive integer. n In page 184, a table of the developments of powers of a binomial is given. These may be immediately deduced from the series we have just proved. Thus, to develop, or expand (a + x), we have— = a3 + 8a2x+28aox2 + 56a3x3 +70a1xa + 56a3x3 +28a2x2 + 8ax2 +x3. The student will observe that the coefficients of a'x and of ax' are the same, as also of ax2 and a2x6 of a3x3 and a3x5. And, in general, if we write the series, whether we begin from a or from x. we get the same coefficients. Thus 28+8x7a+28x6a2 + 56x5a3 + 70x1a1 + 56x3α5 + 28x2α6 +8xa2 + a3 a3 +8ax + 28aox2 + 56a3x3 +70a1xa + 56a3x5 + 28a2x2 + 8ax2 + x3. It can easily be seen that this must be the case, for the former of these two is (x+a), and the latter (a + x), which are clearly the same thing. This consideration greatly facilitates our expansion of a binomial. Thus— |