and its multiplier; and is the product of the 1 1 third term and its multiplier; and so on, to the term whose number is n, whose product by its multiplier is 1 1 n(n + 1) * (n + 1)(n+2) Hence, by adding the preceding products, and erasing the terms that destroy each other, we shall have twice the sum of the series expressed by 1.2 (n + 1)(n+2) = 1 1 1 1 1 (n + 1)(n + 2); and dividing by 2, we get the series 2 3.4 3.4.5 2(n + 1)(n + 2) required. + + Ρ for the sum of n terms of the series, as to stand for a positive integer, such that p + 1 equals the number of factors in the denominator of each of the fractional terms of the series. Here we must multiply each term of the series by p; and we must put p = (p+1) - 1 for the multiplier of the first term of the series; and put p = (p + 2) — 2 for the multiplier of the second term; also, p= (p+3)-3 is the multiplier of the third term, and so on, to p= (p+ n) — n, which is the multiplier of the term whose number is n. If we denote the sum of the n terms of the series by S, then (as before) we shall have S = By putting 2 under the form 2 = m — (m − 2), and dividing mm-1) by m- 2), using m for the first term of the divisor, we get (m m (m — 1 — (m — 1) (m − 2) the remainder; so that (m—1)(m—2) 2 If we now put 2 = (m − 1) — (m − 3), and divide (m − 1) (m-2) by (m-1) — (m − 3), using (m-1) for the first — term of the divisor, we get, as before, m(m − 1) = (m − 1) + (m − 2) + (m − 3) + etc. (1), which will clearly never terminate when m is not a positive integer. If in any calculation we meet with the infinite series (m − 1) + (m − 2) + (m − 3) + etc., we may evidently put for it; because the series is the expansion of Here we put 3 = m — (m − 3), and divide m(m—1)(m—2) by m - (m· 3), using m for the first term of the divisor, (m − 2) (m − 3) (m — 4 (m − 2) (m − 3) + (m-1)-(m-4) ; and in the divi sor (m-1) · 1) — (m — 4), if we change m into m-1, we have (m2)-- (m — 5) = (m − 1) (m-4), and then (m—2)(m—3)(m—4) (m—3)(m—4)(m—5) (m-2)-(m-5) (m—3)—(m—6) and so on. Hence, = = (m−3)(m—4) + m(m − 1)(m − 2) 3 − 3) + (m − 3) (m −4) + etc., (1), which will never terminate if m is not a positive integer. If we meet with the infinite series (m − 1) (m − 2) + (m − 2) (m −3) + etc., in m(m-1) (m2) any calculation, we may evidently put for it. 3 Universally, if n is any positive integer, and we meet with the infinite series (m1)(m2) (m3) × .... x (m-n) +(m − 2)(m − 3) (m −4) × . . . . × (m − n − 1) + (m − 3) (1), for the series, because the series is the expansion of m(m — 1) (m — 2) × . . . . × (m — n) OF INTEGERS; THEIR DIVISORS, FORMS, ETC. (1.) WHEN an integral number is such that it can not be produced by the multiplication of any whole numbers except unity and itself, it is called a prime number. Thus, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, etc., are prime numbers. (2.) Any whole number which can be produced by the multiplication of whole numbers, which are different from unity and the number itself, is called a composite number; and the number is said to be a multiple of any one of its factors. Thus, 4, 6, 8, 9, 10, 12, 14, 15, etc., are composite numbers; and 4 = 2 × 2 is a multiple of 2, and 6 = 2 × 3 is a multiple of 2 or 3, and so on. (3.) Any divisor of an integral number, including unity and the number itself among the divisors, is called a measure of the number. Thus 1 is a measure of 1, 1 and 2 are measures of 2, 1 and 3 are measures of 3; 1, 2, 4 are measures of 4, and so on. (4.) If we include unity among the factors of any integral number, and exclude the number itself, or do not consider the number as a factor of itself, then any integral divisor of the number is often called an aliquot part of the number. Thus 1 is the half of 2, because 2 contains 1 twice; also 2 is the third part of 6, because 6 contains 2, 3 times, and so on. (5.) Any two or more integers or integral quantities that have one or more common factors are said to have any one of the common factors for a common measure or divisor, and the product of all the common factors is called their greatest common measure or divisor. (6.) If unity or one is the greatest common divisor, the numbers or quantities are said to be prime to each other. Thus, 2 and 3 are measures and divisors of 6, 18, and 30, and 6 is the greatest common measure or divisor of the same numbers; also 8 and 15 are prime to each other, since 1 is their greatest common divisor. (7.) Any integer or integral quantity that can be divided by two or more integers or integral quantities, is called their common multiple; and the least integer or integral quantity that can be divided by two or more integers or integral quantities, is called their least common multiple. Thus, since 45 can be divided by 3 and 5, 45 is a common multiple of 3 and 5; but the least integer that can be divided. by 3 and 5 is 15, so that 15 is the least common multiple of 3 and 5. (8.) Any quantity (whether it is integral or not) when mul tiplied by an integer, gives a product that is often called a multiple of the quantity. Thus, 7a, 3b, 2c are called multiples of a, b, c, without reference to the natures of a, b, c. (9.) We will now proceed to establish the following important proposition. If a and b denote any two whole numbers, each greater than unity, and p any prime number which does not divide either a or b, then the product ab of a and b is not divisible by p. For if we take either of the numbers as a, then if a is greater than p, we may put a = mp + a1, supposing m to be the quotient of the division of a by p, and a' the remainder, which by the nature of division is less than the divisor p. Hence we have ab = mpb+ a1b, and if ab is divisible by p, its equal mpb + a'b must be divisible by p; and since mpb is a multiple of p, it is of course divisible by p, consequently a'b must be divisible by p. Hence, in ab we may suppose a to be less than p; and since p is a prime number, it will not be divisible by a without a remainder; hence, if we put m1 for the quotient of the division of p by a, and a1 for the remainder, we shall have p = m'a + a1, and of course pb = m'ab + a1b; then, since pb is divisible by p, its equal m'abab must be divisible by p; consequently, if ab is divisible by p, a'b must be divisible by p. In like manner, if we divide p by a1, and use a" for the remainder, we may show that a"b must be divisible by p when a'b is divisible by p, and so on. III Hence, if ab is divisible by p, then must each of the products a'b, a"b, aшb, a1b, and so on, be divisible by p. But since a1 is less than p, and a", the remainder of the division of p by a', is less than a1; and in the same way aTM less than a", a' less than aTM, and so on, it is evident that we shall soon have one of these integers equal to unity; for there can not be an infinite number of different positive integers, as a, a, a", a, etc., each less than p. Consequently, if in ab we have a = 1, we must have abb, divisible by p, which is against the hypothesis, since b is not divisible by p; hence ab is not divisible by p, as was to be shown. (10.) It follows from what has been done, that if p divides the product ab, and does not divide a, that it must of neces |