Page:EB1911 - Volume 14.djvu/576

was first given by Brook Taylor in his Methodus Incrementorum (1717) as a corollary to a theorem concerning finite differences. Taylor gave the expression for ƒ(x + z) in terms of ƒ(x), ƒ′(x), . . . as an infinite series proceeding by powers of z. His notation was that appropriate to the method of fluxions which he used. This rule for expressing a function as an infinite series is known as Taylor’s theorem. The relation (i.), in which the remainder after n terms is put in evidence, was first obtained by Lagrange (1797). Another form of the remainder was given by Cauchy (1823) viz.,

${\frac {(b-a)^{n}}{(n-1)!}}(1-\theta )^{n-1}f^{n}\left\{a+\theta (b-a)\right\}$.

The conditions of validity of Taylor’s expansion in an infinite series have been investigated very completely by A. Pringsheim (Math. Ann. Bd. xliv., 1894). It is not sufficient that the function and all its differential coefficients should be finite at x = a; there must be a neighbourhood of a within which Cauchy’s form of the remainder tends to zero as n increases (cf. Function).

An example of the necessity of this condition is afforded by the function ƒ(x) which is given by the equation

$f(x)={\frac {1}{1+x^{2}}}+\sum _{n=1}^{n=\infty }{\frac {(-1)^{n}}{n!}}{\frac {1}{1+3^{2n}x^{2}}}$.

(i.)

The sum of the series

$f(0)+xf^{\prime }(0)+{\frac {x^{2}}{2!}}f^{\prime \prime }(0)+\ldots$

(ii.)

is the same as that of the series

$e^{-1}-x^{2}e^{-3^{2}}+x^{4}e^{-3^{4}}-\ldots$

It is easy to prove that this is less than e⁻¹ when x lies between 0 and 1, and also that ƒ(x) is greater than e⁻¹ when x = 1/√3. Hence the sum of the series (i.) is not equal to the sum of the series (ii.).

The particular case of Taylor’s theorem in which a = 0 is often called Maclaurin’s theorem, because it was first explicitly stated by Colin Maclaurin in his Treatise of Fluxions (1742). Maclaurin like Taylor worked exclusively with the fluxional calculus.

Examples of expansions in series had been known for some time. The series for log (1 + x) was obtained by Nicolaus Mercator (1668) by expanding (1 + x)⁻¹ by the method of algebraic division, and integrating the series term by term. He regarded his result as a “quadrature of the hyperbola.” Expansions in
power series. Newton (1669) obtained the expansion of sin⁻¹x by expanding (1 − x²)^−1/2 by the binomial theorem and integrating the series term by term. James Gregory (1671) gave the series for tan⁻¹x. Newton also obtained the series for sin x, cos x, and e^x by reversion of series (1669). The symbol e for the base of the Napierian logarithms was introduced by Euler (1739). All these series can be obtained at once by Taylor’s theorem. James Gregory found also the first few terms of the series for tan x and sec x; the terms of these series may be found successively by Taylor’s theorem, but the numerical coefficient of the general term cannot be obtained in this way.

Taylor’s theorem for the expansion of a function in a power series was the basis of Lagrange’s theory of functions, and it is fundamental also in the theory of analytic functions of a complex variable as developed later by Karl Weierstrass. It has also numerous applications to problems of maxima and minima and to analytical geometry. These matters are treated in the appropriate articles.

The forms of the coefficients in the series for tan x and sec x can be expressed most simply in terms of a set of numbers introduced by James Bernoulli in his treatise on probability entitled Ars Conjectandi (1713). These numbers B₁, B₂, . . . called Bernoulli’s numbers, are the coefficients so denoted in the formula

${\frac {1}{e^{x}-1}}=1-{\frac {x}{2}}+{\frac {{\text{B}}_{1}}{2!}}x^{2}-{\frac {{\text{B}}_{2}}{4!}}x^{4}+{\frac {{\text{B}}_{3}}{6!}}x^{6}-\ldots ,$

and they are connected with the sums of powers of the reciprocals of the natural numbers by equations of the type

${\text{B}}_{n}={\frac {(2n)!}{2^{2n-1}\pi ^{2n}}}\left({\frac {1}{1^{2n}}}+{\frac {1}{2^{2n}}}+{\frac {1}{3^{2n}}}+\ldots \right).$

The function

$x^{m}-{\frac {m}{2}}x^{m-1}+{\frac {m\cdot m-1}{2!}}B_{1}x^{m-2}-\ldots$

has been called Bernoulli’s function of the mth order by J. L. Raabe (Crelle’s J. f. Math. Bd. xlii., 1851). Bernoulli’s numbers and functions are of especial importance in the calculus of finite differences (see the article by D. Seliwanoff in Ency. d. math. Wiss. Bd. i., E., 1901).

When x is given in terms of y by means of a power series of the form

$x=y\left({\text{C}}_{0}+{\text{C}}_{1}y+{\text{C}}_{2}y^{2}+\ldots \right)\quad \left({\text{C}}_{0}\neq 0\right)=yf_{0}(y),{\mbox{ say,}}$

there arises the problem of expressing y as a power series in x. This problem is that of reversion of series. It can be shown that provided the absolute value of x is not too great,

$y={\frac {x}{f_{0}(0)}}+\sum _{n=2}^{N=\infty }\left\lbrack {\frac {x^{n}}{n!}}\cdot {\frac {d^{n-1}}{dy^{n-1}}}{\frac {1}{\left\{f_{0}(y)\right\}^{n}}}\right\rbrack _{y=0}$

To this problem is reducible that of expanding y in powers of x when x and y are connected by an equation of the form

$y=a+xf(y),\,$

for which problem Lagrange (1770) obtained the formula

$y=a+xf(a)+\sum _{n=2}^{N=\infty }\left\lbrack {\frac {x^{n}}{n!}}\cdot {\frac {d^{n-1}}{da^{n-1}}}\left\{f(a)\right\}^{n}\right\rbrack .$

For the history of the problem and the generalizations of Lagrange’s result reference may be made to O. Stolz, Grundzüge d. Diff. u. Int. Rechnung, T. 2 (Leipzig, 1896).

Fig. 10.

38. An important application of the theorem of intermediate value and its generalization can be made to the problem of evaluating certain limits. If two functions φ(x) and ψ(x) both vanish at x = a, the fraction φ(x)/ψ(x) may have a finite limit at a. This limit is described as the limit of an Indeterminate
forms. “indeterminate form.” Such indeterminate forms were considered first by de l’Hospital (1696) to whom the problem of evaluating the limit presented itself in the form of tracing the curve y = φ(x)/ψ(x) near the ordinate x = a, when the curves y = φ(x) and y = ψ(x) both cross the axis of x at the same point as this ordinate. In fig. 10 PA and QA represent short arcs of the curves φ, ψ, chosen so that P and Q have the same abscissa. The value of the ordinate of the corresponding point R of the compound curve is given by the ratio of the ordinates PM, QM. De l’Hospital treated PM and QM as “infinitesimal,” so that the equations PM:AM =φ’(a) and QM:AM = ψ′(a) could be assumed to hold, and he arrived at the result that the “true value” of φ(a)/ψ(a) is φ′(a)/ψ′(a). It can be proved rigorously that, if ψ′(x) does not vanish at x = a, while φ(a) = 0 and ψ(a) = 0, then

$\lim _{x=a}{\frac {\phi (x)}{\psi (x)}}={\frac {\phi ^{\prime }(x)}{\psi ^{\prime }(x)}}.$

It can be proved further if that φ^m(x) and ψⁿ(x) are the differential coefficients of lowest order of φ(x) and ψ(x) which do not vanish at x = a, and if m = n, then

$\lim _{x=a}{\frac {\phi (x)}{\psi (x)}}={\frac {\phi ^{n}(x)}{\psi ^{n}(x)}}.$

If m>n the limit is zero; but if m<n the function represented by the quotient φ(x)/ψ(x) “becomes infinite” at x = a. If the value of the function at x = a is not assigned by the definition of the function, the function does not exist at x = a, and the meaning of the statement that it “becomes infinite” is that it has no finite limit. The statement does not mean that the function has a value which we call infinity. There is no such value (see Function).

Such indeterminate forms as that described above are said to be of the form 0/0. Other indeterminate forms are presented in the form 0 × ∞, or 1^∞, or ∞/∞, or ∞ − ∞. The most notable of the forms 1^∞ is lim._x=0(1 + x)^1/x, which is e. The case in which φ(x) and ψ(x) both tend to become infinite at x = a is reducible to the case in which both the functions tend to become infinite when x is increased indefinitely. If φ′(x) and ψ′(x) have determinate finite limits when x is increased indefinitely, while φ(x) and ψ(x) are determinately (positively or negatively) infinite, we have the result expressed by the equation

$\lim _{x=\infty }{\frac {\phi (x)}{\psi (x)}}={\frac {\lim _{x=\infty }\phi ^{\prime }(x)}{\lim _{x=\infty }\psi ^{\prime }(x)}}.$

For the meaning of the statement that φ(x) and ψ(x) are determinately infinite reference may be made to the article Function. The evaluation of forms of the type ∞/∞ leads to a scale of increasing “infinities,” each being infinite in comparison with the preceding. Such a scale is

$\log x,\ldots x,x^{2},\ldots x^{n},\ldots e^{x},\ldots x^{x};$

each of the limits expressed by such forms as lim._x=∞ φ(x)/ψ(x), where φ(x) precedes ψ(x) in the scale, is zero. The construction of such scales, along with the problem of constructing a complete scale was discussed in numerous writings by Paul du Bois-Reymond (see in particular, Math. Ann. Bd. xi., 1877). For the general problem of indeterminate forms reference may be made to the article by A. Pringsheim in Ency. d. math. Wiss. Bd. ii., A. 1 (1899). Forms of the type 0/0 presented themselves to early writers on analytical geometry in connexion with the determination of the tangents at a double point of a curve; forms of the type ∞/∞ presented themselves in like manner in connexion with the determination of asymptotes of curves. The evaluation of limits has innumerable applications in all parts of analysis. Cauchy’s Analyse algébrique (1821) was an epoch-making treatise on limits.

If a function φ(x) becomes infinite at x = a, and another function ψ(x) also becomes infinite at x = a in such a way that φ(x)/ψ(x) has a finite limit C, we say that φ(x) and ψ(x) become “infinite of the same order.” We may write φ(x) = Cψ(x) + φ₁(x), where lim._x=aφ₁(x)/ψ(x) = 0, and thus φ₁(x) is of a lower order than φ(x); it may be finite or infinite at x = a. If it is finite, we describe Cψ(x)