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Let us suppose the several derived functions of f'(x+h-2), up to the nth inclusive, to be finite and continuous for all employed values of their subject-variables; then, by a series of successive integrations by parts, we have

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and replacing the left-hand member in terms of its equivalent from (40), we have

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2n-1

hn-i ... +7*-!(7) 1.2.3... (n-1) +, "F"(x+h^2)1.2.-1)(42)

nima zh-ldz which is Taylor's Theorem ; and in which the equivalence of the two members of the equation is perfect, and without any indeterminateness. The remainder is given by the definite integral

21-1 dz

; if this can be determined in a finite form, the expansion of f(x+h) is completely exhibited ; and if the remainder is zero, when n = oo, it must be neglected when the number of the terms of the series is infinite. This will of course always be the case, when the series is convergent. In Vol. I, Art. 134, (16), the remainder of Taylor's series is expressed in the form o f " (x +0h), where 0 is a positive proper

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fraction. The equivalence of these two forms of remainders is easily demonstrated. For by the theorem contained in (228), Art. 116, we have 2-1 dz

2-1 dz | F"(x+h-2) 1.2.3... (n-1) =

11 = F"(x + Oh)). 1.2.3.in-1).

az h*
= F* (+ Oh) 1 2 3 n '

(43) where e denotes a positive proper fraction; but as we have no means generally for determining its value, the definite integral is the more exact expression of the remainder.

192.] Maclaurin's Theorem for the development of a function of x may also be demonstrated in a similar manner; and its remainder may be expressed as a definite integral.

Let F'(x-2) be a function of z which is finite and continuous for all employed values of its subject-variables, the value of 2 ranging from 0 to x; then [*r(x-2)dz = [ -=(x-2)]*** = F(x) — F(0).

(44) And suppose also that all the derived functions of f'(x-2) up to the nth are finite and continuous for all employed values of its subject-variables, and for all values of 2 between 0 and x; then by integration by parts we have

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and replacing the left-hand member from (44), we have

F" (x - 2) 1.2..(n-1)

21-1 pr

2*-dz ... + FN

1.2.3... (n-1) which is Maclaurin's series; and in which the equivalence of the two members is perfect, and without any indeterminateness.

· (n-1)} (46)

THE DEV

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193.) THE DEVELOPMENT OF SERIES BY INTEGRATION. 253 The remainder of the series is given by the definite integral

29-1dz

** ; and if this definite integral can be evaluated, the expansion of F(x) is completely exhibited ; and if it = 0, when n = oo, it must be omitted when the number of terms of the series is infinite. This is the case when the series is convergent.

The remainder of Maclaurin's series, as it is given in (18), Art. 134, Vol. I. is f" (0x) ; and this is equivalent to the preceding definite integral; for by (228), Art. 116,

2-1 dz
5. **" (2—2) 1.2.3... (n-1) = F" (0x)], 1.2.3... (n-1)

X) 1.2.3 ... n'

. : (47) where @ denotes a positive proper fraction; but as we have generally no means of determining it, the definite integral is the more exact expression of the remainder.

SECTION 3.-The Development of Series by means of

Single Definite Integration. 193.] In discussing the several methods which have been employed for the approximate determination of the value of a definite integral, when that value cannot be found directly and in finite terms, we shewed how in many cases the element-function or part of the element-function might be expanded into a converging series; whereby it would consist of a series of terms, of each of which the definite integral could be determined; and thus the value of the whole definite integral could be, at least approximately, found. This process we explained and illustrated in Arts. 119-121; and have applied in Art. 157 to the rectification of an ellipse of small eccentricity.

Also in exhibiting the uses of the Gamma-function and of its allied integrals in Arts. 143–145 we were incidentally led to the sum of certain series of a complex character under the form of definite integrals.

Now the correctness of these processes is based on the following theorem. If an element-function can be expressed in a convergent

S

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series, so that the series is equivalent to the element-function and can be used instead of it, then the definite integral of the element is equal to that of the series; the limits being the same in both cases, and neither the element-function nor any term of the series being infinite or discontinuous for any value of the variable within the range of integration. Consequently, if the definite integral of all the terms of the series can be found, the sum of them is the value of the original definite integral.

In the preceding cases this method has been employed for the purpose of approximating to the value of a definite integral, when that value cannot otherwise be found. Here however we propose to apply the method to integrals, the values of which can otherwise be determined ; so that the definite integral will be expressed as a series; and thus either the definite integral may be considered as the sum of a series, or the series may be considered as the expansion of the definite integral. Consequently the process is that of summation or of expansion according to the point of view whence it is considered.

194.] The following are examples of the process :

Ex. 1. Since the following series found by the binomial theorem is convergent for all values of x greater than – 1,

1
1 +ã
-= 1-2 +22 - 23 + ... ;
tine

(48)

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tan-1

(51) which series is convergent for all values of x less than 1, and is useful for the calculation of T. Also if x = 1,

1 1 1 1

+57 +ği # 1 1 1

(53) :: ğ = 1.3 + 5.7 + 9.11 +....

(52)

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In the case where x is greater than 1, that is, when tan-1x is intermediate to and , the following series is more convergent; since

1 1 1 + ?

more con

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1

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1
.. tan-1% = 5-o+ 3.73 – 525 T.

(54) which series is however the same as (51), when x is replaced

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Ex. 3. Again, since

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sin-1x = x+1 28 4 1.3 203 1.3.5 x?

2+2 3+24 5 + 2.4.6 9 +.... (55) Hence, if x = 1,

-1 11, 1.3 11.3.5 1...
= 1+2 3 + 24 5 + 24.6 7 7

+ .... (56) This series however converges too slowly to be of use for the calculation of t. It is to be observed too that although we have taken the superior limit to be 1, which makes the elementfunction infinite, yet the series is correct; because the value of the element-function corresponding to the superior limit is not included in the definite integral. Again, in (55), let x = 5; then 1 1 1 1 1.3 1 .1.3.5

1

5 7 6 = 2 + 2 3(2)3 + 2.4 5(2)5 + 2.4.6 7/2j7 + .. (67)

mcluded in the del corresponding to th: because the value of

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