Problem 51 - Thomas Calculus

Question

The Taylor series generated by \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) is \(\sum_{n=0}^{\infty} a_{n} x^{n}\) A function defined by a power series \(\Sigma_{n=0}^{\infty} a_{n} x^{n}\) with a radius of convergence \(R>0\) has a Taylor series that converges to the function at every point of \((-R, R) .\) Show this by showing that the Taylor series generated by \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) is the series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) itself. \begin{equation} \begin{array}{c}{\text { An immediate consequence of this is that series like }} \\ {x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}+\cdots} \\ {\text {and}}\end{array} \end{equation} \begin{equation} x^{2} e^{x}=x^{2}+x^{3}+\frac{x^{4}}{2 !}+\frac{x^{5}}{3 !}+\cdots \end{equation} obtained by multiplying Taylor series by powers of \(x,\) as well as series obtained by integration and differentiation of convergent power series, are themselves the Taylor series generated by the functions they represent.

Step-by-Step Solution

Answer

The Taylor series \( \sum_{n=0}^{\infty} a_n x^n \) is itself, with convergence on \((-R, R)\).

Step 1: Understanding the Power Series

A power series is an infinite series of the form \( f(x) = \sum_{n=0}^{\infty} a_n x^n \), where \( \{a_n\} \) are coefficients and \( x \) is a variable. The series has a radius of convergence \( R \), meaning it converges for \( |x| < R \).

Step 2: Formalizing the Taylor Series

The Taylor series of a function \( f(x) \) at \( x = 0 \) can be expressed as \( \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \), where \( f^{(n)}(0) \) is the \( n \)-th derivative of \( f \) evaluated at \( x = 0 \).

Step 3: Comparing Power Series to Taylor Series

In the context of this exercise, the function \( f(x) \) defined by \( \sum_{n=0}^{\infty} a_n x^n \) means \( f^{(n)}(0) = n! \cdot a_n \) so that \( \frac{f^{(n)}(0)}{n!} = a_n \). Thus, the Taylor series for \( f(x) \) at \( x = 0 \) is \( \sum_{n=0}^{\infty} a_n x^n \), which matches the original power series.

Step 4: Verifying Convergence

Since the original series \( \sum_{n=0}^{\infty} a_n x^n \) has a radius of convergence \( R > 0 \), it converges to \( f(x) \) at any point \( x \) in \( (-R, R) \). This ensures that the Taylor series equals the original function \( f(x) \) in this interval.

Step 5: Generalizing Beyond Basic Series

The result implies that series formed by multiplying the power series by powers of \( x \), or by integrating or differentiating the series, remain the Taylor series of the respective functions. Examples include functions like \( x \sin x \) and \( x^2 e^x \), which can be derived from the known Taylor series of \( \sin x \) and \( e^x \), by these operations.

Key Concepts

Power Series

A power series is an essential mathematical tool that allows us to write functions as an infinite sum of terms. Each term in a power series is a product of a coefficient \(a_n\) and a power of \(x\): \(\sum_{n=0}^{\infty} a_n x^n\). This setup makes it easy to evaluate, approximate, and manipulate functions.

• Each coefficient \(a_n\) plays a crucial role, altering the series' behavior and dictating its convergence properties.
• The term \(x^n\) ensures the series has the flexibility to represent a wide range of functions.

Power series can approximate complex functions, giving us a means to analyze them precisely. They are a cornerstone in calculus and analysis because they transform complex functions into simple polynomial-like forms, making them easier to work with.

Radius of Convergence

The radius of convergence is a crucial concept when dealing with power series. It indicates the interval around a central point where the series converges to a function. For a power series \(f(x) = \sum_{n=0}^{\infty} a_n x^n\), the series converges when \(|x| < R\), where \(R\) is the radius of convergence.

• If \(R\) is large, the series converges for a wide interval, allowing it to represent the function across a vast domain.
• Conversely, if \(R\) is small, the series has a limited range of convergence.

The radius of convergence is determined through various tests, such as the ratio test or root test. Understanding this concept ensures that the function's representation through a power series is valid in the specified interval.

Function Representation

Representing functions using power series or Taylor series allows for a more straightforward analysis and manipulation of complex functions. In the context of this exercise, a function \(f(x)\) is expressed as \(\sum_{n=0}^{\infty} a_n x^n\), serving as its own Taylor series under certain conditions.

• For the Taylor series, the coefficients are derived from the function's derivatives evaluated at a point, usually \(x = 0\).
• When the original power series coincides with the Taylor series, it means the function's natural form is already optimized for analysis.
• This equivalence allows us to treat the convergence properties of the power series confidently and use it to understand the function's behavior through calculus operations, like multiplication, differentiation, or integration.

By using power series for function representation, complex operations become simple algebraic manipulations, opening pathways to solving mathematical problems with precision and ease.