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Ex. 7. Find the sum of the series
12. 2? + 22.2 + 32.28 + &c. Sum to n terms = n2. 2" + En*. 2" 2" 2
4 =n'. 2" +
2 1 2-1 (2 - 1) = 2" (2no – 4n +6} + C. The method just given may be generalized to apply to all functions of the form Uz. $ (), where $ () is rational and integral, and Uz is a function such that we know the value of Aug for all integral values of n. In this case we have (comp. Ex. 3, p. 20)
Euz $(2)=(EE' — 1)2 47$ (Qc) =(AE' + A)-2478(x) (E' and A' being supposed to operate on $ and E and A on Uz alone)
+4*U. APP (oc — 3) - &c. ... (11), dropping the accents as no longer necessary.
Ex. 8. A good example of the use of the above formula is got by taking Ug = sin (ax + b). From (17), page 8, we get easily
sin An sin (ax + b) =
Let us take then the series whose nth term is
(n - 7) sin (an +b);
the sum of n terms will be
(n – 7) sin (an+b) + (n 7) sin (an + 6)
a + sin (an +6
2 = (n - 7) sin (an +b) + (n − 8).
– (a +T)} + C.
6th. Miscellaneous Forms. When a function proposed for integration cannot be referred to any of the preceding forms, it will be proper to divine if possible the form of its integral from general knowledge of the effect of the operation A, and to determine the constants by comparing the difference of the conjectured integral with the function proposed. Thus since
Aa*b (a) = a*y (x), where f(a) = a (x + 1) - (x), it is evident that if $(a) be a rational fraction f (ac) will also be such. Hence if we had to integrate a function of the form af of (ac), 7 (2) being a rational fraction, it would be proper to try first the hypothesis that the integral was of the form a*$ (Qc), $ (2) being a rational fraction the constitution of which would be suggested by that of t (a).
Thus also, since Asin? $ (), A tan? $ (a), &c., are of the respective forms sing (a), tant (X), &c., f (w) being an algebraic function when $ (Qc) is such, and, in the case of tan? (2), rational if (a) be so, it is usually not difficult to conjecture what must be the forms, if finite forms exist, of
Esin f (x), Etan? if (oc), &c., z (@) being still supposed algebraic,
The above observations may be generalized. The operation denoted by A does not change or annul the functional
characteristics of the subject to which it is applied. It does not convert transcendental into algebraic functions, or one species of transcendental functions into another.
And thus, in the inverse procedure of integration, the limits of conjecture are narrowed. In the above respect the operation A is unlike that of differentiation, which involves essentially a procedure to the limit, and in the limit new forms arise.
Instances of the above will be given in the Examples at the end of the chapter, but we subjoin the following by way of illustration.
Ex. 9. To sum, when possible, the series
1'.. 2.. 3*.*
+ &c. to n terms.
n2. an The nth term, represented by Um, being
(n + 1) (n + 2) we have
+Σ (n + 1) (n + 2) (n + 1) (n + 2) Now remembering that the summation has reference to.n, assume
an + 6
a(n+1) +b an+61
n+1) a (oc — 1) n? +(2a + b) (-1)n+(a+b)x - 26
(n+1) (n + 2) That these expressions may agree we must have a (x - 1)=1, (2a+b) (20 – 1) =0, (a + b) x — 26 = 0. Whence we find
n + 2
The proposed series is therefore integrable if x=4*, and we have
1 n 2 Σ
.4" + C. (n+1) (n + 2) 3'n +1 Substituting, determining the constant, and reducing, there results 12.4 22. 42 n24"
3 3. Eis of course, like A, E, and D, an operation capable of repetition and therefore obeying the index-law; E'un being defined as E (Eur). Our symbolical methods will render it an easy matter to obtain expressions for E" (or A") analogous to those already obtained for £, but we shall have to add, as in Integral Calculus, a function of the form
C. + Cz+...+ C, zo-1 (where C., C,, &c. are arbitrary or undetermined constants) instead of the single arbitrary constant which we added in the previous instance. We shall merely give the formula for En analogous to (10) and leave the others as an exercise for the ingenuity of the student. It is
am Σ*a*φ (α)
* The explanation of this peculiarity is very easy:
+ (n+1) (n +2)
n +2 n t and the summation of the above series would require a finite expression for Σ 2 if x had not such a value that the term which occurs in the
- 4x2 (r + 1)th term exactly cancelled the term that occurs in the oth term,
i. e. unless 2=4.
It will be found that the 1st, 3rd, and 5th forms can have their nth Finite Integrals expressed in finite terms, but that the 2nd and 4th only permit of this if n be not too great.
Conditions of extension of direct to inverse forms. Nature
of the arbitrary constants. 4. From the symbolical expression of £ in the forms (D-1-7), and more generally of E" in the form (– 1)", How certain theorems which may be regarded as extensions of some of the results of Chap. II. To comprehend the true nature of these extensions the peculiar interrogative character of the expression (edx 1)-*up must be borne in mind. Any legitimate transformation of this expression by the development of the symbolical factor must be considered, in so far as it consists of direct forms, to be an answer to the question which that expression proposes ; in so far as it consists of inverse forms to be a replacing of that question by others. But the answers will not be of necessity sufficiently general, and the substituted questions if answered in a perfectly unrestricted manner may lead to results which are too general. In the one case we must introduce arbitrary constants, in the other case we must determine the connecting relations among arbitrary constants; in both cases falling back upon our prior knowledge of what the character of the true solution must be. Two examples will suffice for illustration.
Ex. 1. Let us endeavour to deduce symbolically the expression for Euz, given in (3), Art. 1. Now Euz=(E – 1)-? Us
= (E-+E?+ &c.) U,
=u-+U+Us... +&c. Now this is only a particular form of Eur corresponding to a=-00 in (3). To deduce the general form we must add an arbitrary constant, and if to that constant we assign the value
-(, +Waz... + &c.), we obtain the result in question.