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But multiplying (10) by 2*,

2.4.6 ... 2. = 2*x +s

2x = C** ........

(12).

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Therefore, dividing (11) by (12),

3.5.7... (2x - 1) = 2z+***,

2.4.6 ... 2.c whence 3.5.7... (2x - 1)

23
But by Wallis's theorem, w being infinite,

2.4.6... (2. — 2) (2x)
3.5.7 ...

(2xC – 1) whence by division

w (2.c)

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.. C = log (27). And now, substituting this value in (9) and determining

Ux, we find

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1

1

If we develope the factor 12x 36023 +&c.

in descending powers of x, we find

1 1 1.2.3... =w (272).2cm €-*(1+ +

12. * 288.262

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Hence for very large values of w we may assume

1. 2. 3...= V(213) () (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 r (2+1) when a is large.

It will be seen that this reduces to the preceding example when x is integral; it has been chosen to illustrate our mode of determining C. Exactly as in the preceding case we obtain

B,

B; log uz = C + C + (x + a 2 +

&c. 1. 2x 3.4x3 5.6.

(16), 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 d’ log Ta 1 1

1
+
+

+ &c. ad inf.
dx2

22 (3 + 1)2 (2c + 2)

= 0) when x is infinite.

But from (16) we obtain, when x is infinite, da log T (+1) _d2C

which is therefore zero when daca

dx2' x is infinite.

Now C is a periodic quantity going through its course of

d? values as x increases by unity-hence

is equally pedaca

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But C remains unchanged when & 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 i was an integer, i.e. C= log V2.

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1

1 2n
= 0 -

+
(2n-1) 22n-1 20,21

12.0c2nti
2n (2n + 1) (2n + 2)
+

720x

&c.

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 "uz in a series proceeding by the differential coefficients of Uz

Since &=(co-1)+; ::: E"=(P-1)", and the problem reduces to that of expanding (e' — 1)in ascending powers of t; or, in other words, to expanding (€ – 1)" in positive integral powers of t.

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Multiply both sides by n-1 and let w.=in-1vn, and the equation becomes

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Ultimately we obtain (writing n-1 for n)

d

d In-1 {€ – 1}"=(-1)"-1 tn-1 ti

at

...( + 1){c+–13-.........(17).

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

Ex. To develope & in terms of D.

From (17),

12(e-13=(+ 2) (+1) [4–11"
- (+3+18

2) {6 – 1+4,6+44€ + &c.} suppose,

Bor+1

where Agr=0 for all values of r and Agr+1 = (-1)"

| 2r + 2

B, F. D.

7

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+ £ ?(+2) (r+1) Ar+2+3 (n+1) Ar1 + 2A,] t*.

r=1

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6. PROP. III. To develope Suur in a series, proceeding by successive differential coefficients of Ux_?

n 2

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2

S*u,= D^{1+0:09) + c()*+&c.} v._....-(19)*:

4 dx3

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

1 B, dur + B, du
Us

- &c. - C... (20),
2

2 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 En 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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