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But by Wallis's theorem, x being infinite,

2.4.6... (2x-2) √(2x) =33 =
3.5.7... (2x-1)

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And now, substituting this value in (9) and determining

Ux, we find

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Hence for very large values of x we may assume

1.2.3......x = √(2πx) (~)*.

(15),

the ratio of the two members tending to unity as a tends to infinity. And speaking generally it is with the ratios, not the actual values of functions of large numbers, that we are concerned.

Ex. 5. To find an approximate value of г (x+1) when x is large.

It will be seen that this reduces to the preceding example when a is integral; it has been chosen to illustrate our mode of determining C.

Exactly as in the preceding case we obtain

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but we can draw no conclusion as to the value of C from the value it bore in (9), nor would any number of special determinations of its value enable us to draw any conclusions as to its general value. But it can be proved (Todhunter's Int. Cal. 3rd Ed. p. 254) that

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But from (16) we obtain, when x is infinite,

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Now C is a periodic quantity going through its course of

values as x increases by unity—hence

d2Ŭ
dx2

is equally pe

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But C remains unchanged when x is increased by unity; therefore A = 0, and C is therefore an absolute constant, and therefore has the value found for it in Ex. 4 when x was an integer, i. e. C'= log √2π.

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For each particular value of n the constant C might be determined approximately as in Ex. 3, but its general expression will be found in Art. 3, Ch. VI.

5. PROP. II. To develope Eu, in a series proceeding by the differential coefficients of ux.

Since

Σ=(eo−1) ́1; .. Σ" = (e”—1)TM”,

and the problem reduces to that of expanding (-1) in ascending powers of t; or, in other words, to expanding th in positive integral powers of t.

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Multiply both sides by n-1 and let wn-1, and the equation becomes

Wn+1

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Wn-1

− (de + n) w. = ( 2 + n) (de + n − 1) w.,

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dt

wn

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dt

dt

Ultimately we obtain (writing n-1 for n)

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By means of this formula we can obtain developed expressions for 2, 3, &c. with great readiness in terms of the coefficients in the expansion of Σ, i.e. in terms of Bernoulli's numbers.

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where A,,=0 for all values of r and A2+1 = (-1)"

B. F. D.

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Hence

++(24+34 ̧ − 1)

r=x

+ Σ [(r• + 2) (r + 1) Ar+2 + 3 (r + 1) Ar+1 + 2A,] t、

r=1

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6. PROP. III. To develope "u, in a series, proceeding by successive differential coefficients of u2_n.

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It must be mentioned that the Summation-formula of Art. 2 (which is due to Maclaurin†) is quite as applicable in the form

3

Ju_dx = Σu + 1 u, -B, du,+B, du; - &c. - C'... (20),

2

Их

2 dx 4dx3

to the evaluation of integrals by reducing it to a summation, as it is, in its original form, to the summation of series by reducing it to an integration. It is thus a substitute for (27), page 54.

* This remarkably symmetrical expression for Zn is due to Spitzer (Grunert, Archiv. XXIV. 97).

+ Tract on Fluxions, 672. Euler gives it also (Trans. St Petersburg, 1769), and it is often ascribed to him.

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