We shall now commence the first of these divisions.

2. PROP. I. To develope Eu, in a series proceeding by the differential coefficients of u

d

Since Σu ̧= (e« — 1) ̄1u, we must expand (ez — 1)~1 in

d ascending powers of and the form of the expansion will dx' be determined by that of the function (e-1). For simplicity we will first deduce a few terms of the expansion and examine somewhat its general form, leaving fuller investigations to the next Chapter.

The function (e* — 1) ̃1 is not at once suitable for expansion by Maclaurin's Theorem, since it contains a negative power

of t; we shall therefore expand

t

either by Maclaurin's

Theorem or by actual division and divide the result by t,

The term

2

2 12 720

may be shewn to be the only term in the

expansion involving an odd power of t. For

which does not change when t is changed into t, and therefore can contain, on expansion, even powers of t alone..

From these results we may conclude that the development of (e* — 1)-1 will assume the form

It is however customary to express this development in the somewhat more arbitrary form

The quantities B1, B,, &c. are called Bernoulli's numbers, and will form the subject of the major part of the next chapter.

Or, actually calculating a few of the coefficients,

The following table contains the values of the first ten of Bernoulli's numbers,

* Attention has been directed (Differential Equations, p. 376) to the interrogative character of inverse forms such as

d

( − 1) 1.

The object of a theorem of transformation like the above is, strictly speaking, to determine a function of x such that if we perform upon it the cor

d

responding direct operation (in the above case this is ex-1) the result will be u. To the inquiry what that function is, a legitimate transformation will necessarily give a correct but not necessarily the most general answer. Thus C in the second member of (6) is, from the mode of its introduction, the constant of ordinary integration; but for the most general expression of Eu, C ought to be a periodical quantity, subject only to the condition of resuming the same value for values of x differing by unity. In the applications to which we shall proceed the values of x involved will be integral, so that it will suffice to regard C as a simple constant. Still it is important that the true relation of the two members of the equation (6) should be understood.

It

It will be noted that they are ultimately divergent. will seldom however be necessary to carry the series for Zu further than is done in (7), and it will be shewn that the employment of its convergent portion is sufficient.

Applications.

3. The general expression for Eu, in (7), Art. 2, gives us at once the integral of any rational and entire function of x.

which at once enables us to connect Bernoulli's numbers with the coefficients of the powers of x in the expression for

But the theorem is of chief importance when finite summation is impossible.

The value of C must be determined by the particular conditions of the problem. Thus suppose it required to determine an approximate value of the series

6

Let x = ∞, then the first member is equal to by a known theorem, while the second member reduces to C. Hence

and if x be large a few terms of the series in the second member will suffice.

4. When the sum of the series ad inf. is unknown, or is known to be infinite, we may approximately determine C by giving to x some value which will enable us to compare the expression for Eu, in which the constant is involved, with the actual value of Σu, obtained from the given series by addition of its terms.

Hence, writing for log, 10 its value 2.302585, we have approximately C=577215. Therefore

Ex. 4. Required an approximate value for 1.2.3... x.

.. logu, = C+ (x + 1) log x − x + 12 - &c. .... (9).

To determine C, suppose a very large and tending to become infinite, then

log (1. 2. 3... x) = C + (x + 1) log x − x,