Question

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

Short answer

Taylor polynomials and Taylor series are both methods to approximate a function centered at point a. The relation between them is that the Taylor polynomial of degree n represents the partial sum of the first n+1 terms of the Taylor series. As the degree of the Taylor polynomial increases, the approximation of the function gets closer to the actual function and approaches the Taylor series.

Step-by-step solution

Define Taylor polynomial and Taylor series centered at a

A Taylor polynomial of degree n for a function \(f\) centered at \(a\) is a polynomial that approximates the function near the point \(a\). It is given by the formula: \[P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k\] Where \(f^{(k)}(a)\) is the k-th derivative of the function evaluated at the point \(a\). A Taylor series for a function \(f\) centered at \(a\) is an infinite series that represents the function as a sum of terms, where each term involves \((x-a)^k\) and the corresponding derivatives of the function evaluated at \(a\). It is given by the formula: \[T(x)=\sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{k!}(x-a)^k\]

Relate Taylor polynomial to Taylor series

Both Taylor polynomials and Taylor series involve terms with the same structure: \(\frac{f^{(k)}(a)}{k!}(x-a)^k\). The main difference between them is that a Taylor polynomial has a finite number of terms (up to the degree n), while a Taylor series has an infinite number of terms. To relate the Taylor polynomials to the Taylor series, we can think of the Taylor polynomial of degree n as the partial sum of the first n+1 terms of the Taylor series: \[P_n(x) = \sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = T_n(x)\] Where \(T_n(x)\) represents the partial sum of the first n+1 terms of the Taylor series \(T(x)\).

Conclusion

In conclusion, the Taylor polynomials for a function \(f\) centered at \(a\) are related to the Taylor series of the function centered at \(a\) by being the partial sums of the Taylor series. The Taylor polynomial of degree n consists of the first n+1 terms of the Taylor series, providing an approximation of the function near the point \(a\). As the degree of the Taylor polynomial increases (n goes to infinity), the Taylor polynomial approaches the Taylor series, and the approximation gets closer to the actual function.

Key concepts

Taylor series

A Taylor series is a powerful mathematical tool used to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This series is particularly useful because it can help approximate complex functions with a series of simpler polynomial terms.
The formula for a Taylor series centered at a point \(a\) is:\[T(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!} (x-a)^k\]Here, each term is built on:

• The \(k\)-th derivative of the function \(f\) evaluated at \(a\), \(f^{(k)}(a)\).
• The factorial of \(k\), represented as \(k!\), which helps scale the impact of higher derivatives.
• The term \((x-a)^k\), marking the point \(a\) where the series is centered.

What makes the Taylor series fascinating is its ability to resemble a given function more closely as more terms from the series are taken into account. When the number of terms approaches infinity, the Taylor series can converge to become the function itself, provided certain criteria are met, such as the function being smooth and infinitely differentiable.

Partial sums

Partial sums in the context of Taylor series are finite sums that approximate the series by adding only a limited number of terms. They represent the sum of the first \(n+1\) terms of the infinite series.
These sums are denoted \(T_n(x)\) and expressed as:\[T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k\]Each partial sum is effectively a Taylor polynomial of degree \(n\), providing an approximation of the actual Taylor series. The more terms included (i.e., the higher the \(n\)), the more accurate the approximation becomes.
This approach allows mathematicians and scientists to achieve practical estimates of functions without resorting to evaluating potentially endless terms.

• Partial sums are used when infinite precision isn't needed.
• They simplify computations while maintaining a reasonable degree of accuracy.

Partial sums are how Taylor polynomials are actually derived from the Taylor series, and they demonstrate the practical application of the series in approximating functions effectively.

Degree of polynomial

The degree of a polynomial is a fundamental concept when working with polynomials in math, and it applies directly to Taylor polynomials as well. In the context of Taylor polynomials and partial sums, the degree refers to the highest power of \((x-a)\) included in the polynomial.
It's given by the number \(n\) in the polynomial expression:\[P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k\]The degree of the polynomial dictates the number of terms in the Taylor polynomial. Specifically, a Taylor polynomial of degree \(n\) contains terms up to \((x-a)^n\). These terms allow the polynomial to approximate a function locally around the point \(a\).
Understanding the degree is vital because:

• The degree directly influences the accuracy of the approximation; higher degrees generally lead to more precise approximations of the function.
• It helps determine computational complexity; higher-degree polynomials require more calculations.

By gradually increasing the degree, one can improve the approximation of the original function offered by the Taylor polynomial, showing how polynomial degree and function approximation are closely linked.