From (3) and (4), by addition and subtraction,

A

2

Ρ

By ascribing to p in succession the values 2, 4, 6,..., we see from (6) that each of the coefficients A, A,, A.,... is equal to zero; and from (5) we obtain

By equating the absolute terms in (2), we obtain

1 = A ̧ + A ̧ + Â ̧+ Â ̧+ + A ̧ ;

1

2

3

......

and by putting n = 1 in equation (1), we have

thus

406.

The result of the preceding article is most conveniently expressed by the formula,

59

n

The quantities B1, B ̧, B ̧‚..... are known as Bernoulli's Numbers; for examples of their application to the summation of other series the advanced student may consult Boole's Finite Differences.

Example. Find the value of 15 +25 + 35 +

+n5.

14. Without assuming the formula, find the sum of the series:

(1) 16+26+36 +............ •+no

(2) 17+27+37+.

+n?.

16. Shew that the coefficient of " in the expansion of

17. If n is a positive integer, find the value of

18. If n is a positive integer greater than 3, shew that

20. Sum to infinity the series whose nth term is

23. Prove that, if a<1, (1+ax)(1 + a3x) (1+α3x)......

24. If A, is the coefficient of x" in the expansion of

25. If n is a multiple of 6, shew that each of the series

27. If P, (nr) (n−r+1) (n−r+2)......(n−r+p−1),

CHAPTER XXX.

THEORY OF NUMBERS.

407. In this chapter we shall use the word number as equivalent in meaning to positive integer.

A number which is not exactly divisible by any number except itself and unity is called a prime number, or a prime; a number which is divisible by other numbers besides itself and unity is called a composite number; thus 53 is a prime number, and 35 is a composite number. Two numbers which have no common factor except unity are said to be prime to each other; thus 24 is prime to 77.

408. We shall make frequent use of the following elementary propositions, some of which arise so naturally out of the definition of a prime that they may be regarded as axioms.

(i) If a number a divides a product be and is prime to one factor b, it must divide the other factor c.

For since a divides bc, every factor of a is found in bc; but since a is prime to b, no factor of a is found in b; therefore all the factors of a are found in c; that is, a divides c.

(ii) If a prime number a divides a product bcd..., it must divide one of the factors of that product; and therefore if a prime number a divides b", where n is any positive integer, it must divide b.

(iii) If a is prime to each of the numbers b and c, it is prime to the product bc. For no factor of a can divide b or c; therefore the product be is not divisible by any factor of a, that is, a is prime to be. Conversely if a is prime to be, it is prime to each of the numbers b and c.

it is

Also if a is prime to each of the numbers b, c, d, prime to the product bcd... ; and conversely if a is prime to any number, it is prime to every factor of that number.