How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?

Verified step by step guidance

1. Recall that the Taylor polynomial of degree n for a function \( f \) centered at \( a \) is given by the formula: \[ P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k \] where \( f^{(k)}(a) \) denotes the \( k \)-th derivative of \( f \) evaluated at \( a \).
2. Understand that the Taylor series of \( f \) centered at \( a \) is the infinite sum of all these terms, expressed as: \[ T(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!} (x - a)^k \] This series represents the function \( f \) as an infinite polynomial expansion around \( a \).
3. Recognize that each Taylor polynomial \( P_n(x) \) is essentially the partial sum of the Taylor series up to degree \( n \). In other words, the Taylor polynomial approximates \( f \) by truncating the infinite series after \( n \) terms.
4. Note that as \( n \) increases, the Taylor polynomial \( P_n(x) \) generally provides a better approximation to \( f(x) \) near the point \( a \), and the Taylor series is the limit of these polynomials as \( n \to \infty \), assuming the series converges to \( f(x) \).
5. Therefore, the Taylor polynomials centered at \( a \) are the finite-degree approximations that build up to the full Taylor series centered at \( a \), which is the infinite-degree polynomial representation of \( f \) around that point.

Key Concepts

Taylor Polynomial

A Taylor polynomial of degree n for a function f centered at a is a finite sum that approximates f near a using derivatives of f at a. It includes terms up to the nth derivative, providing a polynomial approximation that becomes more accurate as n increases.

Taylor Series

The Taylor series of a function f centered at a is an infinite sum of terms derived from the derivatives of f at a. It represents the function as a power series and, if convergent, equals the function within a certain interval around a.

Relationship Between Taylor Polynomials and Taylor Series

Taylor polynomials are partial sums of the Taylor series; each polynomial includes a finite number of terms from the series. As the degree of the polynomial increases, it approaches the Taylor series, which is the limit of these polynomials as the number of terms goes to infinity.