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3. Find an approximate value of

3.5......(2.0 + 1)

2.4......2.C supposing a large but not infinite.

1 4. Find approximately & and obtain an exact for

22 a>

1 2

mula when a is an integral multiple of.

5. Transform the series

1
* +1

1

1

+ 202 + 4

- &c., ca +9

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1

1

+ &c., ac (x + 1) (x + 2) (c + 1) (3 + 2) (+3) into series of a more convergent character, and find an approximate value of the sum of each when x = 5, that is, correct to 6 places of decimals.

6. If x + u 2 + Updc2 + &c. = f (), shew that U7Vx+ U7Vx+2+ &c. = f (1) +f* (1) Av.

f" (1) 1.2

Apvx + &c. and apply this theorem to transform the series

+

Xolm)

+ 2
(x + 1)(-m)

+ 3
(x + 2)(m)

+ &c.

2

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2+1
2 - 3+1

and shew that its sum ad inf. is

9. Shew by the method given in the note to page 72, that

1

In 1" + 2" +....

+ x2
+ x" +

+&c.,
n+1
[2 n - 2

-1

2n+1

xn-1

B,

n

d" where Brys

doc" 1

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[Schlömilch, Grunert x. 342.] 10. Shew that the sum of all the negative powers of all whole numbers (unity being in both cases excluded) is unity;

3 if odd powers are excluded it is

1 11. Expand Σ

in terms of successive differences

(ax + b)" of log (ax + b) and deduce

A? Σ

log sin x + log sin a

&c. 2

3

[Tortolini, v. 281.] 12*. If Sn = Us + Un + Wen + &c., ad inf., shew that

na 1 SM

Ug
Au +

&c.
2
12n

24n

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n - 1

na – 1

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AU

n

r=0

1 13. Find in factorials, and determine to 3 places

3 of decimals the value of the constant when the first term is

1 (31)**

If the Maclaurin Sum-formula had been used, to what degree of accuracy could we have obtained C?

* De Morgan (Dif. Cal. 554). Compare (27), page 54.

14. Shew that

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414

1

1
+ + &c., ad inf.,
et -1

1
C– log x 1 (B.)

(B)
+

28 - &c. 4

2 2 and apply this to the summation of Lambert's* series, viz. 22

1
+ + &c., when x is = nearly. .
1

[Zeitschrift, vi. 407.] 15. Shew that

f(0) + f(1) + ...... ad inf.
1

ett tett f(- kt) f (xt)
еті
ent

2K where K=v-1,

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dt,

and deduce similar formulæ for the sums of the series

f (0) - f (1) + f (2) - &c.,

f (1) + f (3) + f (5) + &c. Find an analogous expression for the sum of the last mentioned to n terms.

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e-ot

.c e(1-2)t-e-(17 – «) atdt
ent

a' + t if x lie between it and

[Schlömilch, Crelle XLII. 130.]

T.

* On the application of the Maclaurin Sum-formula to this important series see also Curtze (Annali Math. 1. 285).

CHAPTER VI.

رارا

BERNOULLI'S NUMBERS, AND FACTORIAL COEFFICIENTS.

1. THE celebrated series of numbers which we are about to notice were first discovered by James Bernoulli. They first presented themselves as connected with the coefficients of powers of x in the expression for the sum of the nth powers of the natural numbers, which we know is

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C+

204+1

n-1

2 B.

B.
-
+ +

n (n − 1) (n − 2) an-3
n+1
2 12

- &c....... (1),
or rather as the coefficient of x in the successive expressions
when n was an even integer, and De Moivre pointed out that
by taking this between limits 1 and 0 we obtain the formula
1 1

n (12 – 1) (n − 2)
1" =
+ +
B,

B,+ ...... (2), 2 2

4

n

1

n+1

from which the numbers can be easily calculated in succession by taking n=2, 4, ......

After the discovery of the Euler-Maclaurin formula [(6), page 90] the coefficients were shewn to be those of 1

from the application of it to Eekst, which gives - 1

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-1

which gives
1 1 1 B,

B.
+ "h hs + &c....

&c........... .(4). en-1 h

2'12 4 2. Many other important expansions can be obtained by consideration of this identity. Thus, for h write 207–1; then, since 1 1 (220v-1+1

1 1

cot A e2017 2 28-1-1

27-1

2 we at once obtain

1 B,
cot A 2- B42498 – &c.

.(5). \2

A
Again cosec 0 = cot cot 0,

2
1
B

B. .:. cosec 0 2 0 +2 (28 – 1) S. AS + &c. ... (6). e 12

14 Similarly from cot 0 - 2 cot 20 = tan o we obtain

22 (22 – 1) 24 (24 – 1) tan o B0+ B.A% + &c.

(75... 12

| 4 3. An expression for the values of the numbers of Bernoulli

d can be obtained from (5). For cot A (log sin 6) and

1

2 29 –

do

log sin 0 =10340(1-) --

= log 8+log (1-%) +&e. ot 0–3-2 (1-9)**(1-2)

*-&c. -*++5+ &c.}

... cotA=

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