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We shall now commence the first of these divisions.
2. PROP. I. To develope. Eu, in a series proceeding by the differential coefficients of uz Since Lus=(e* – 1)", we must expand (€* – 194 in
d ascending powers of and the form of the expansion will
dac' be determined by that of the function (é' - 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 (€ – 1) is not at once suitable for expansion by Maclaurin's Theorem, since it contains a negative power of t; we shall therefore expand either by Maclaurin's
€ -1 Theorem or by actual division and divide the result by t,
t The term
2 expansion involving an odd power of t. For
- may be shewn to be the only term in the
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)-- will assume the form
It is however customary to express this development in the somewhat more arbitrary form
1 1 B. B, B (6 - 1)
t t + 5 t – &C.......(5).
2'12 4° 76 The quantities B,, B,, &c. are called Bernoulli's numbers, and will form the subject of the major part of the next chapter. Hience we find
1 B, du, B, du Euz= = C + dc
+ &c.......(6). 2x
(2 dx [4 das Or, actually calculating a few of the coefficients,
1 1 duz 1 duz Eur= =
1 dur 30240 dəs
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
(edx – 1)-lur. The object of a theorem of transformation like the above is, strictly speak. ing, to determine a function of x such that if we perform upon it the corresponding direct operation (in the above case this is edt – 1) the result will
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 a 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 will be noted that they are ultimately divergent. It will seldom however be necessary to carry the series for Eur 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.
More generally, making U, = 2c" we get
1 nB, Exc"
n (n − 1) (n − 2) B
L4 - &c.
which at once enables us to connect Bernoulli's numbers with the coefficients of the powers of w in the expression for
But the theorem is of chief importance when finite summation is impossible.
1 Ex. 2. Thus making Ug =
we have cತೆ ?
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 1 1 1
(x - 1) Now by what precedes, 1 1 1
1 1 1 1 1:+*+&+...
&c. (wc — 1)
72 Let x = 00, then the first member is equal to by a known
6 theorem, while the second member reduces to C. Hence
1 т? 1 1 1 1 1+ ... +
+ &c. (vc — 1) 6. 3C 2.c 6x3 3006 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 w some value which will enable us to compare the expression for Eux, in which the constant is involved, with the actual value of Eur obtained from the given series by addition of its terms.
1 1 1 Ex. 3. Let the given series be 1 +5+ +
Representing this series by us, we have
1 Ur=- + Σ
Hence, writing for loge 10 its value 2:302585, we have approximately 0= -577215. Therefore
1 Un=-577215 + log 2 +
Ex. 4. Required an approximate value for 1.2.3 ... x.
To determine C, suppose x very large and tending to become infinite, then
log (1.2. 3 ... ) = C+(2+) log æ – ,