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1.3.5 3.5.7 5.7.9
-1.3.5.cos 0 + 3.5.7. cos 20 +5.7.9. cos 30+..

1+ 2a cos 0 + 3a cos 20 + 4a$ cos 30 + ... 2. The successive orders of figurate numbers are defined by this ;—that the acth term of any order is equal to the sum of the first x terms of the order next preceding, while the terms of the first order are each equal to unity. Shew that the oth term of the nth order is

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

1 3. If & Uz denote the sum of the first n terms of the series Up, U7, U4, &c. shew that

1 1 A Ao

2 2

8 and apply this to find the sum of the series



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1.3.5 +5.7.9 +9.11.13 + &c.

4. Expand E$ (@) cos mx in a series of differences of $ (oc).

5. Find in what cases, when U, is one of the five forms given as integrable in the present Chapter, we can find the sum of n terms of the series

U– u,+u, -Ug+&c., and construct the suitable formulæ in each case.



6. Sum the following series to n terms :


1 1
sin 0
sin 20

sin 40

cos O. cos 20 cos 20. cos 30




7. Shew that cot"? (P + qn +rn*) is integrable in finite terms whenever

q? – ga = 4(pr:- 1). Obtain E tan

) Σ

Σ 1+ n(n 1) c'


n(n+1) 8. It is always possible to assign such values to s, real or imaginary, that the function

(a + Bac + yac+ ... + vac").se

• Ux+m-1 shall be integrable in finite terms; a, B... v being any constants and ux = ax + b.

(Herschel's Examples of Finite Differences, p. 47.) 9. Shew that


U + U, cos 20 + U, cos 40+ &c. =

2 4 sino

sin +

sin 38 - &c.
8 sin
16 sina 8

32 sin


cos 20


an 10. If Au, = Uzth - Up and 1= shew that

a" - 1 1. + λΔu,+λ'Δ'u + &c. +λ"Δ"u,

=a**{(ak – 1) Ea*uz +2 "Earth An+14;}. Find the sum of n terms of the series whose nth terms are

(a+n-1)"x"-? and (a +n - 1)(m).0"-1. 11. Prove the theorem

n(n+1) Σ*, = uΣ"υ, - ηΔυ, Σ***υχα +

A'u, Entsy - &c.

1.2 12. If $ (Qc) = v. + v,x + vac? + &c., shew that U7V + 2,0,8 + u,vac* + &c. = u$ (Qc) + x^(2). Au,

xca +1.29"(). A’u, + &c.;

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and if $(w) = v. +0,2 +000(2) + &c., then
U,%+ 2,00 + 0,9, 20(?) + &c. = 2.0 (c) + XA$(x). Av.

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(Guderman, Crelle, VII. 306.) 13. Sum to infinity the series

2(2-1) z(2 – 1) (2 – 2) O" + 1". .

+ 3".

t... x (x - 1) 2 (2 — 1) (x - 2)

+ 2".

14. If (X) = v. + 0,2 + 0,20* + &c., shew that Ay Upaca + Qr+nUp+n@c + Op+2nUr+2nXr+2n + &c.


{E. [$(am)] u. + [a**** $'(ac)] Au, . +&c.},

where a is an nth root of unity.

15. If 1" + 2n + + m= S, and m(m + 1) =p, shew that Sn=p* f (p) or (2m + 1)pf (P), according as n is odd or even.

(Nouvelles Annales, X. 199.)



1. It has been seen that the finite summation of series depends upon our ability to express in finite algebraical terms the result of the operation performed upon the general term of the series. When such finite expression is beyond our powers, theorems of approximation must be employed. And the constitution of the symbol as expressed by the equation

&=(@D – 1)2... (1) renders the deduction and the application of such theorems easy.

Speaking generally these theorems are dependent upon the development of the symbol in ascending powers of D.

But another method, also of great use, is one in which we expand in terms of the successive differences of some important factor of the general term, i.e. in ascending powers of Ă, where A is considered as operating on one factor alone of the general term, and is no longer the inverse of the we are trying to perform*.

* Let us compare these methods of procedure with those adopted in the Integral Calculus. If (x) dx cannot be obtained in finite terms it is usual either

(1) To expand • (20) in a series proceeding by powers of x and to integrate each term separately;

(2) To develope (x) dx by Bernoulli's Theorem (i.e. by repeated integration by parts) in a series proceeding by successive diferential coefficients of some factor of the general term; or

As our results are no longer exact it becomes a matter of the greatest importance to determine how far they differ from the exact results, or, in other words, the degree of approximation attained. But this is usually a difficult task, and in order to lessen the difficulty of the subject to the student, we shall separate such investigations from those which first give us the expansions. The order in which we shall treat the subject will therefore be as follows:

1. We shall. obtain symbolical expansions for £, £, &c. (Chapters V. and vi.)

II. We shall examine the general question of Convergency and Divergency of Series, to ascertain if we may assume the arithmetical equivalence of the results of performing on uz the operations that we have just found to be symbolically equivalent. (Ch. VII.)

III. Finding that many of our results do not stand the test we shall proceed to find the exact theorems corresponding to them, i.e. to find expressions for the remainder after n terms, and thus we shall reestablish the approximateness of these results. (Ch. VIII.)

(3) To develope (c) dx in a series proceeding by successive differences of px by aid of Laplace's formula for Mechanical Quadrature [(27) page 54], which may be written thus: for (x) dx=C+2°(2) 1, 10(-x)+, 24(x) – &c. ... (2).

24 We should therefore expect to find in the Sum-Calculus the corresponding methods, viz. :

(1) To expand Ux in a series proceeding by factorials, and to sum each term separately;

(2) To develope Euz in a series proceeding by successive differences of some factor of the general term;

(3) To develope Eux in a series proceeding by successive differential coefficients of Ux

Of these (3) and (2) are those mentioned in the text; (1) is not of much use since the cases in which it can be applied are very few, and no theorems of great generality have been found to enable us to obtain the expansion necessary. Besides the resulting series will usually be highly divergent unless the factorials are inverse ones, i.e. have negative indices, so that the results will not be suitable for giving the approximate values we seek. We shall, however, give some account later on of the results that have been obtained by this method.

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