24. Shew that, in the expansion of (1+x+x2 +......x")", where n is a positive integer, the coefficient of terms equidistant from the beginning and the end are equal. 25. If α, α, α......be the coefficients in the expansion of (1+x+x2)" in ascending powers of x, prove that 2 2 2 · + ( − 1 )”− 1 a„2 = {} {an − (− 1)” an3}. 26. If (1 + x + x2)′′ = α。 + α ̧ x + α ̧ x2 + ɑ2 prove that 27. Shew that, in the expansion of (a,+a+a+...+a ̧)", where n is a whole number less than r, the coefficient of any term in which none of the quantities a1, a2, &c. appears more than once is n! ...... 28. Shew that, if the quantities 1 + ∞, 1 + x + x3, ....... (1 + x + x2 + +x") be multiplied together, the coefficients of terms equidistant from the beginning and end will be equal; and that the sum of all the odd coefficients will be equal to the sum of all the even, each being (n + 1) ! 29. Shew that the coefficient of x" in the expansion of (1 + x + x2)" is 30. Shew that 18 can be made up of 8 odd numbers in 792 different ways, where repetitions are allowed and the order of addition is taken into account. CHAPTER XXI. CONVERGENCY AND DIVERGENCY OF SERIES. 260. A series is a succession of quantities which are formed in order according to some definite law. When a series terminates after a certain number of terms it is said to be a finite series, and when there is an endless succession of terms the series is said to be infinite. We have already found that when the common ratio of a geometrical progression is numerically less than unity the sum of n terms will not increase indefinitely, but that the sum will become more and more nearly equal to a fixed finite quantity as n is increased without limit. Thus the sum of an infinite series is not in all cases infinitely great. When the sum of the first n terms of a series tends to a finite limit S, so that the sum can, by sufficiently increasing n, be made to differ from S by less than any assignable quantity, however small, the series is said to be convergent, and S is called its sum. Thus 1+1+1+}+..... is a convergent series whose sum is 2. When the sum of the first n terms of a series increases numerically without limit as n is increased indefinitely, the series is said to be divergent. Thus 1+ 2+ 3+ 4+... is a divergent series. When the sum of the n first terms of a series does not increase indefinitely as n is increased without limit, and yet does not approach to any determinate limit, the series is neither convergent nor divergent. Such a series is sometimes called an indeterminate or a neutral series*. For example, the series 1-1+1-1+... is an indeterminate series, for the sum of n terms is 1 or 0 according as n is odd or even. It is clear that a series whose terms are all of the same sign cannot be indeterminate, but must either be convergent or divergent. For unless the sum of n terms increases without limit as n is increased without limit, there must be some finite limit which the sum can never exceed, but to which it approaches indefinitely near. 261. If each term of a series be finite, and all the terms have the same sign, the series must be divergent. For, if each term be not less than a, the sum of n terms will be not less than na, and na can be made greater than any finite quantity, however large, by sufficiently increasing n. 262. The successive terms of a series will be denoted by u,, u, u,...; and, since it is impossible to write down all the terms of an infinite series, it is necessary to know how to express the general term, u,, in terms of n. ; The sum of the n first terms will be denoted by U and the sum of the whole series, supposed convergent, in which case alone it has a sum, will be denoted by U. U = u1 + u2 + uz + ... + U1 + Um + 1+..., 1 2 n Thus ...... 263. In order that the series u,, U „ U z U ş Un+1 &c. may be convergent it is by definition necessary and sufficient that the sum * These series are however called divergent series by Cauchy, Bertrand, Laurent and others. should converge indefinitely to some finite limit U as n is indefinitely increased. Hence Un Unt Un+ &c. ... must differ from U, and therefore from one another, by quantities which diminish indefinitely as n is increased without limit. Hence, in order that a series may be convergent, the (n+1)th term must decrease indefinitely as n is increased indefinitely, and also the sum of any number of terms beginning at the (n+1)th must become less than any assignable quantity, however small, when n is indefinitely increased. gent, although the nth term diminishes indefinitely as n is increased indefinitely; for the sum of n terms beginning at the (n + 1)th is Indefinitely 1 1 2n' 1 which is greater than xn, that is, greater 2n 264. We shall for the present consider series in which all the terms have the same sign; and as it is clear that the convergency or divergency of such a series does not depend on whether the signs are all positive or all negative, we shall consider all the signs to be positive. The convergency or divergency of series can generally be determined by means of the following theorems. 265. Theorem I. A series is convergent if all its terms are less than the corresponding terms of a second series which is known to be convergent. Then, since u<v, for all-values of r, it follows that U is less than V. Hence, as V is finite, U must also be finite this proves the theorem, for a series must be convergent when its sum is finite and all the terms have the same sign. It can be proved in a similar manner that a series is divergent if all its terms are greater than the corresponding terms of a divergent series. The terms of the series are less than the terms of the series 1 metrical progression whose common ratio is which is therefore is convergent if a, b and x are all positive, and a <b. The terms of the given series are less than the corresponding terms of the series [since ra + x a+ x b+x (a + x)2, (a + x)3 < if r > 1, a, b and x being positive and b> a]. rb+x The latter series is convergent, and therefore also the given series. To ensure the convergency of the first series it is not necessary that all its terms should be less than the corresponding terms of the second series, it will be sufficient if all the terms except a finite number of them |