91Ó°ÊÓ

Taylor polynomials are finite-degree polynomial approximations of a function \(f\) centered at a point \(a\). The \(n\)th-degree Taylor polynomial \(P_n(x)\) of the function \(f\) centered at \(a\) is given by: \[P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k\] where \(f^{(k)}(a)\) is the \(k\)th derivative of \(f\) evaluated at \(a\).

A Taylor series is an infinite series representation of a function \(f\) centered at a point \(a\). The Taylor series \(S(x)\) of the function \(f\) centered at \(a\) is given by: \[S(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!}(x-a)^k\]

The relationship between the Taylor polynomials and Taylor series of a function \(f\) centered at \(a\) can be seen when comparing their general forms: \[P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k\] \[S(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!}(x-a)^k\] A Taylor series is an infinite sum of the terms that make up Taylor polynomials. As n approaches infinity, the Taylor polynomial \(P_n(x)\) converges to the Taylor series \(S(x)\). In other words, the Taylor polynomial is a truncated version of the Taylor series, and the Taylor series is the limit of the Taylor polynomials as the degree approaches infinity. In conclusion, the Taylor polynomials for a function \(f\) centered at \(a\) are related to the Taylor series for the function \(f\) centered at \(a\) as finite-degree approximations. As the degree of the Taylor polynomial increases, the approximation becomes more accurate, and it converges to the Taylor series representation.