Home page for accesible maths Math 101 Chapter 4: Taylor series and complex numbers 4.2 Taylor series and complex numbers chapter contents 4.4 Taylor’s polynomials

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

4.3 Polynomial approximation

A polynomial function has the shape

p(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dots+a_{n}x^{n},

where the $a_{j}\,$ are the coefficients, and $x\,$ the variable. When $a_{n}\neq 0\,$ , we call $a_{n}x^{n}\,$ the leading term, and $n\,$ the degree. Polynomials are useful in calculations since they can be evaluated by addition and multiplication.

We shall show that suitable functions may be evaluated by the use of series such as

\sin x=x-{{x^{3}}\over{3!}}+{{x^{5}}\over{5!}}-\dots.

By taking sufficiently many terms in the series we can obtain a good approximation to the function. How does one choose the coefficients in this series? After considering many special cases, Maclaurin and Taylor found a general formula.