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This last investigation only applies when z—w is an integer, or in other words when the series would terminate. It is evident that if it were not so we could not draw conclusions as to the magnitude of F(z) – F() from the successive differences as we do above. The form of the periodical constant would affect F(z) – F(x) without affecting the other side of the equation.
Remainder in the Maclaurin Sum-formula. 6. In finding the remainder in the Maclaurin sum-formula we shall take it in the slightly modified form obtained by
both sides. It then becomes
1 dur B, Δυο
Δ + Δ &c. ....... (9), da
2- dac 1.2 dxs
By Taylor's Theorem we have (Todhunter's Int. Cal. Ch. iv.)
where P = u2n+1
Substitute in (10) and the coefficient of u is
This must vanish through the identity expressed in (10). Our symbolical work is the demonstration of this.
We shall now shew that $ (2n, z) does not change sign between the limits of the integral, remains positive or negative as m is even or odd, and has but one maximum (or minimum) value in each case. We see from (11) that $ (r, 2) vanishes when z=1, as it also does when z= = 0.
7. Assume the above to hold good for some value of n, say an even one, so that $ (2n, z) is positive between 0 and 1, has but one maximum and vanishes at the limits. Add thereto A2 (which is negative) and integrate and we obtain $ (2n +1, ). Now this vanishes at both limits, and therefore its differential coefficient (2n, 2) +42n must vanish at some point between them. Now this last is negative at each limit and has but one maximum, thus it must vanish twice, -in passing from negative to positive and from positive to negative, --so that • (2n+1, 2) has only one minimum followed by a maximum between 0 and 1, and thus can vanish but once. Adding Agent (which is zero) to it, for the sake of symmetry, and integrating again we obtain $ (2n+2, z). This vånishes also at both limits, and its differential coefficient is, as we have seen, at first negative and then positive, changing sign but once. Thus 0 (2n + 2, ) has but one maximum and remains positive, which was what we sought to prove. Continuing thus, the theorem is proved for all subsequent values of n, if it be true for any particular one; and as it is true for $(2, 2) or it is generally true.
8. Since $ (2n, z) retains its sign between the limits Ran
• (2n, z) dz, 0 <1>0
2n +1 EU
Now perform £ on both sides of (9) and write (u_dx
Let M be the greatest value irrespective of sign that
duckn has between the limits of summation, x and x + m suppose. Then Eumite must lie between the limits 1 mM.
9. Other conclusions may be drawn relative to the size of the error when other facts are known about the behaviour of Uge and its differential coefficients between the limits. For instance, if u, keeps its sign throughout, we may take 0 instead of - mM as one of the limits. The sign of the error will therefore be that of (-1)" M, and, should w
keep the same sign as un between the limits, the error made by taking one term more of the series will have the same sign as (-1)+1 M, i.e. the true value will lie between them. This is obviously the case in the series at the top of page 101, hence that series (without any remainder-term) is alternately greater and less than the true value of the function.
10. If urs
20+1 retain its sign between the limits in (10) we have
φ (2η, θ) , A <1. Now it can be shewn that (2n, 0) is never greater numerically than – 2 An; hence the correction is never so much as twice the next term of the series were it continued instead of being closed by the remainder-term. Thus, wherever we stop, the error is less than the last term, provided that the differential coefficient that appears therein either constantly increases or constantly decreases between the limits taken. This condition is satisfied in all the important
1 series of the form E The series to which they lead on
ac" application of the Maclaurin sum-formula all converge for a time and then diverge very rapidly. In spite of this divergence we see that they are admirably adapted to give us approximate values of the sums in question, for we have but to keep the convergent portion and then know that our error is less than the last term we have kept; and by artifices such as that exemplified on page 100, this can be made as small as we like.
11. Several solutions have been given of the problem of finding the remainder after any number of terms of the Maclaurin sum-formula.
The one in the text is by Malmstén, and the proof given was suggested by that in a paper by him in Crelle (xxxv. 55). It has been chosen because the limits of the error thus obtained are perfectly general and depend on no property of Uz or the differential coefficients thereof, save that such as appear must vary continuously between the limits. The idea of the method used in this very valuable paper was taken from Jacobi, who used it in a paper on the same subject (Crelle, XII. 263), entitled De usu legitimo formulæ summatorice Maclauriance. Malmstén’s paper contains many other noteworthy results, and in various cases gives narrower limits to the error than those obtained by other
processes, while at the same time they are not too complicated. But the whole paper is full of misprints, so that it is better to read an article of Schlömilch (Zeitschrift, 1. 192), in which he embodies the important part of Malmstén's article, greatly adding to its value by shewing the connection between the remainder and Bernoulli's Function of which we have spoken in Art. 14, page 116. The paper is written with even more than his usual ability, and is to be highly recommended to those who wish further information on the subject.
12. The chief credit of putting the Maclaurin sum-formula on a proper footing, and saving the results it gives from the suspicion under which they must lie as being derived from diverging series, is due to Poisson. In å paper on the numerical calculation of Definite Integrals (Mémoires de 1'Académie, 1823, page 571) he starts from an expansion by Fourier's Theorem, and obtains for the remainder an expression of the form
and he then investigates the limits between which this will lie. The investigation is continued by Raabe (Crelle, XVIII. 75), and the practical use of the results in the calculation of Definite Integrals examined and estimated, and modifications suitable for the purpose obtained.
A method of obtaining the supplementary term which possesses many advantages is based on the formula
1 poo F (C+ ) – F (3C – K2) F(x) +
e272 - 1 where Kav-1. On this see a paper by Genocchi (Tortolini, Ann. Series, 1. Vol. 111.), which also contains plentiful references to earlier papers on the subject. Tortolini in the next volume of the same Journal extends it to En. See also Schlömilch (Grunert Archiv, xii. 130).
13. The investigation which appeared in the first edition of this book is subjoined here (Art. 16). The editor thinks that the fundamental assumption, viz. that the remainder may be considered as being equal to
cannot be held to be legitimate, since the series which the latter represents may be and often is divergent. For the conditions under which the series itself would be convergent, see a paper by Genocchi (Tortolini, Ann. Series, 1. Vol. VI.) containing references to some results from Cauchy on the same subject. There is a very ingenious proof of the formula itself by integration by parts, in the Cambridge Mathematical Journal, by J. W. L. Glaisher, wherein the remainder is found as well as the series, and Schlömilch (Zeitschrift, 11. 289) has obtained them by a method of great generality, of which he takes this and the Generalized Taylor's Theorem as examples.
14. By far the most important case of summation is that which occurs in the calculation of log In and its differential coefficients. For special examinations of the approximations in this case we may refer to papers by Lipschitz (Crelle, LVI. 11), Bauer (Crelle, LVII. 256), Raabe (Crelle, xxv. 146, and XXVIII. 10). It must be remembered that there is nothing to prevent there being two semi-convergent expansions of the same function of totally different forms, so that the discrepancy noticed by Guderman (Crelle, xxix. 209) in two
1 expansions for log In, one of which contains a term in and the other does not, does not justify the conclusion that one must be false.
15. The investigation into the complete form of the Generalized Taylor's Theorem is derived from a paper by Crelle in the twenty-second volume of his Journal. Other papers may be found in Liouville, 1845, page 379, (or Grunert Archiv, viii. 166), Grunert, xiv. 337, and Zeitschrift, 11. 269. The convergence and supplementary term of the expansion in inverse factorials (Stirling's Theorem) have also been investigated by Dietrich (Crelle, Lix. 163).
The degree of approximation given by transformations of slowly converg. ing series has been arrived at by very elementary work by Poncelet (Crelle, XIII. 1), but the results scarcely belong to this chapter.