91Ó°ÊÓ

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 \(\sum_{n=0}^{\infty} a_{n} x^{n}\) 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. An immediate consequence of this is that series like $$x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}+\cdots$$ and $$x^{2} e^{x}=x^{2}+x^{3}+\frac{x^{4}}{2 !}+\frac{x^{5}}{3 !}+\cdots.$$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

01

Understand the Function Definition

The function is defined by a power series: \( f(x) = \sum_{n=0}^{\infty} a_{n} x^{n} \). This means the function is represented as an infinite sum of terms \( a_{n} x^{n} \).

02

Identify the Taylor Series

The Taylor series of a function \( f(x) \) centered at \( x = 0 \) is exactly its power series representation: \( \sum_{n=0}^{\infty} a_{n} x^{n} \), assuming the series is centered at 0.

03

Verify Radius of Convergence

The radius of convergence \( R \) is the range within which the power series converges, meaning the series sums to the function \( f(x) \) for each \( x \) within \( (-R, R) \).

04

Show Series Equals Taylor Series

For a function defined by a power series \( \sum_{n=0}^{\infty} a_{n} x^{n} \), since the power series is already centered at 0, the Taylor series for \( f(x) \) is the same series. Therefore, \( \sum_{n=0}^{\infty} a_{n} x^{n} \) is the Taylor series.

05

Note the Immediate Consequence

Since multiplication by powers of \( x \), and integration and differentiation of the series maintain convergence within \( (-R, R) \), the resultant series are still the Taylor series representing the modified functions (e.g., \( x \sin x \), \( x^2 e^x \)).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Series

A power series is essentially an infinite sum of terms that involve powers of a variable, typically expressed as \( \sum_{n=0}^{\infty} a_{n} x^{n} \). Each term in the series is composed of a coefficient \( a_{n} \) and a power of \( x \), making power series a versatile tool for function representation.

Power series serve as a fundamental way to represent functions due to their simplicity and flexibility, and are often used at the core of calculus and analysis.

• The coefficients \( a_{n} \) are numbers that determine the influence of each \( x^{n} \) term in the series.
• Power series can model a wide variety of functions, often making them easier to approximate near a certain point.
• For power series centered at 0, the series provides a straightforward extension to Taylor series.

One crucial aspect of power series is their utility in defining functions even when finding a function's direct formula is challenging. Thanks to the algebra of power series, functions can be manipulated by differentiating, integrating, or multiplying the series to explore and define new functional relationships.

Radius of Convergence

The concept of the radius of convergence \( R \) is essential in the study of power series. This determines the interval \((-R, R)\) over which the power series converges and accurately represents the function.

• The radius of convergence is a non-negative value, allowing us to understand and define where the series behaves predictably.
• Inside this convergence interval, the power series sums up to provide the actual value of the function represented.
• Outside the radius, the series may diverge, which means it won't provide the desired function value.

Understanding the radius of convergence helps in practical applications, ensuring that mathematical models and predictions are accurate. For solving problems, the radius is often determined using tests like the ratio or root test, providing analytical control over the infinite nature of power series.

Function Representation

The beauty of power series lies in their ability to represent complex functions in a manageable way. Specifically, when a function is expressed as a sum \( f(x) = \sum_{n=0}^{\infty} a_{n} x^{n} \), this not only serves as the power series but also as the Taylor series for the function, making it double as effective for analysis.

• When centered at 0, the power series is also known as a Maclaurin series.
• Functions can be expanded into series to simplify various operations like integration and differentiation.
• Through connection with Taylor series, power series offer insight into function behavior over an interval.

Taylor series allow for precise approximation of functions near the center point and emphasize the overlap of the power series methodology for practical problem-solving. By being adept at translating functions into and from power series, we easily leverage these expansions for computation and theoretical examination in diverse fields of mathematics.