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13. Prove that

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1.

* - &c. =0. 1. 2 1.2.3.4 14. Express the sums of the powers of numbers less than n and prime to it in series involving Bernoulli's numbers.

[Thacker, Nouvelles Annales, x. 324.]

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15. If
€ -1

++
{(1 + E)" E"} P.= 0,

(E+)** P,=0,

(1+2)P, = log (1+x) 16. Shew, in the notation of the last question, that (

+ &c.) On+1 2

4 5

E*+(E – 1) (E – 2) P,=(-1)"

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17. Shew that

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227

1 1 where

Br=1- + - &c.

32r and hence find the sum of such a series in terms of Bernoulli's numbers.

[Dienger, Crelle, XXXIV. 91.] 18. Shew that*

1 1 1

+ &c.

33

32'

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1 1

575 1- +

&c. 35

1536 Many similar summations will be found in a paper by Tchebechef [Liouville, XVI. 337].

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19. If S" = 1"+2" + ... + 2c", shew that

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20. In the notation of Art. 14, page 116, shew that F(x)

1 is a rational and integral function of

and cannot con

2 tain both odd and even powers of the same.

[Bertrand, Diff. Cal. 350.] 21. Shew how the method contained in the note on page 109 could be made to give us the actual values of the numbers of Bernoulli by application of Staudt's Theorem.

22. Apply the formula,

006. + min = (1 + 1)(1--) (1+;-)....

to demonstrate (12), page 110.

[Stern, Crelle, XXVI. 88.]

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22n (221 - 1) Ex=F (2n), and

Bon-= F (2n-1) numerically. 2n

[Schlömilch, Crelle, XXXII. 360.] 24. Shew that

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Σ [(m +1) Σ {(m + 2) E (m+p)}]
[Hargreave, Quarterly Journal, VIII. 26.]

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25. *Shew that f(D) 0" = (nF

(D) 0=

0.

dA"-1 Δη *Prove that {0+ (n − n)}"+r expresses the sum of all the

LL homogeneous products of s dimensions which can be formed of the p + 1 consecutive numbers n, n-1, ... n - r. 26. Express galm) x xm in factorials.

[Elphinstone, Quarterly Journal, II, 254.] 27. If log (1+x)= A,x+ 4,22) + 4,21% +&c.,

1 3 1 shew that

4

4 6 28. If K," = number of combinations of m things r together with repetitions,

C," = number of combinations of m things r together without repetitions,

Ammtr then K,"

and C," is obtained by writing –(m+1)

4.

m

for m in the expanded expression for K,".

[Wasmund, Grunert, xxxiv. 440.] 29. Shew that in the notation of Art. 10 C," 0,"+Cj" – &c. = 0,

1 and B, C," B, C," + &c.

2

[Grunert, 1x. 333.] 30. Shew that

n

n-1 n+1

(x - 1)...... (2 m +1)

cos mzdz; and find from this an expression for the coefficients of the powers of w in the expanded factorial x() in the form of a definite integral.

[Grunert, xi. 447.] Jeffery (Quarterly Journal, 1v. 364).

31. Deduce (26) page 115, from (21) page 24.
32. Shew that 0,8 = 3, and C= 85.
33. Shew that if k<n the coefficient of

SC

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(-1)" CM-K is

(m - 1)(x 2)...(3 x)^ in the notation of Art. 10.

[Schlömilch, Grunert, XVIII. 315.] 34. Shew that (with the notation of (21), page 113)

1 An

+(1+

+ Aint

1.2 and find the general formula for r= k.

Shew that

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36. Find expressions for Bernoulli's numbers and Factorial-coefficients in the form of determinants.

[Tortolini, Series II. VII. 19.]

CHAPTER VII.

Read

CONVERGENCY AND DIVERGENCY OF SERIES.

1. A SERIES is said to be convergent or divergent according as the sum of its first n terms approaches or does not approach to a finite limit when n is indefinitely increased.

This definition leads us to distinguish between the convergency of a series and the convergency of the terms of a series. The successive terms of the series

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converge to the limit 0, but it will be shewn that the sum of n of those terms tends to become infinite with n.

1+

+

+ &c.

On the other hand, the geometrical series

1 1 1 1

+ +

2 4 8 16 is convergent both as respects its terms and as respects the sum of its terms.

2. Three cases present themselves. 1st. That in which the terms of a series are all of the same or are ultimately all of the same sign. 2ndly. That in which they are, or ultimately become, alternately positive and negative. 3rdly. That in which they are of variable sign (though not alternately positive and negative) owing to the presence of a periodic quantity as a factor in the general term. The first case we propose, on account of the greater difficulty of its theory, to consider last.

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