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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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16. Shew, in the notation of the last question, that

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E'TM11 ( E — 1) (E − 2) p ̧ − (− 1)" ( — — — " + &c.) 01.

2

17. Shew that

5

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and hence find the sum of such a series in terms of Ber

noulli's numbers.

18. Shew that*

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[Dienger, Crelle, XXXIV. 91.]

1 1
+
33 58

1

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35 55

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

19. If S1"+2" + ... + x", shew that

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[Eisenlohe, Crelle, XXVIII. 193.]

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

1

is a rational and integral function of x 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,

х

1

1

cos 27+ sin 2 = (1 + x) ( 1 − x) (1 + )..............

4

to demonstrate (12), page 110.

х

)(

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=

2n

В2-1= кF' (2n − 1) numerically.

2n-1

[Schlömilch, Crelle, XXXII. 360.]

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

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*Prove that

{0+ (nr)}str expresses the sum of all the

homogeneous products of s dimensions which can be formed of the r+1 consecutive numbers n, n-1, ... n — r.

26. Express (m) x xn) in factorials.

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[Elphinstone, Quarterly Journal, II, 254.]

27. If log (1+x) = Ã ̧x+ Â ̧ï12) + Ag1x(3) +&c.,

shew that

1 3

4.= {log + log 5}.

A

4 6

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

m

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

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and C is obtained by writing - (m + 1)

m

for m in the expanded expression for K.".

[Wasmund, Grunert, XXXIV. 440.]

29. Shew that in the notation of Art. 10

C” — C,”+ C,” − &c. = 0,

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n 1

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2/m

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x (x-1)...... (x — m + 1) =

π

S

2

2 cos

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

x

[Grunert, XI. 447.]

* Jeffery (Quarterly Journal, Iv. 364).

31. Deduce (26) page 115, from (21) page 24. 32. Shew that C3, and C = 85.

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[Schlömilch, Grunert, XVIII. 315.]

34. Shew that (with the notation of (21), page 113)

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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

1 1 1 1

1+ 2+ 3+ 4+ 5+ &c.

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

On the other hand, the geometrical series

1 1 1 1

1+ + + + + &c.

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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