It is easy to prove that this is less than e−1 when x lies between 0 and 1, and also that f(x) is greater than e−l 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 Expansions in power series. regarded his result as a “quadrature of the hyperbola.” Newton (1669) obtained the expansion of sin−1x by expanding (l − 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 ex 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
| x | = 1 − | x | + | B1 | x2 − | B2 | x4 + | B3 | x6 − ..., |
| ex − 1 | 2 | 2! | 4! | 6! |
and they are connected with the sums of powers of the reciprocals of the natural numbers by equations of the type
| Bn = | (2n)! | ( | 1 | + | 1 | + | 1 | + ... ). |
| 22n−1 π2n | 12n | 22n | 32n |
The function
| xm − | m | xm−1 + | m·m − 1 | B1 xm−2 − ... |
| 2 | 2! |