Maclaurin series

Named after Scottish mathematician Colin Maclaurin (1698-1746), who made extensive use of the series.

Maclaurin series (plural Maclaurin series)

1. (calculus) Any Taylor series that is centred at 0 (i.e., for which the origin is the reference point used to derive the series from its associated function); for a given infinitely differentiable complex function ${\displaystyle \textstyle f}$ , the power series ${\displaystyle \textstyle f(0)+{\frac {f'(0)}{1!}}x+{\frac {f''(0)}{2!}}x^{2}+{\frac {f'''(0)}{3!}}x^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}\,x^{n}}$ .
   • 1953, Raymond Lyttleton, The Stability of Rotating Liquid Masses, Cambridge University Press, 2013, Paperback Edition, page 42,

     Analytically there are, of course, two Jacobi series branching off the Maclaurin series, but they are geometrically and physically identical, and involve only an interchange of a and b.

   • 1995, Ralph P. Boas, Gerald L. Alexanderson (editor), Dale H. Mugler (editor), Lion Hunting and Other Mathematical Pursuits, Mathematical Association of America, page 88,

     If the Maclaurin series of f and g converge for |z| < r and g(z) ≠ 0 for 0 ≤ |z| < r, then if the Maclaurin series for f is divided by the Maclaurin series for g by long division (as if the series were polynomials), the resulting series represents f / g for |z| < r.

   • 1997, Frank Smithies, Cauchy and the Creation of Complex Function Theory, Cambridge University Press, page 203,

     It was almost as a by-product of this work that, in the first Turin memoir, he proved the convergence of the Maclaurin series of a function up to the singularity nearest to the origin (Section 7.5); it was in this context that he created what he called 'calculus of limits', later known as the method of majorants.

• (Taylor series centred at 0): power series, Taylor series