Question

Prove: If $$ f(x)=\sum_{n=0}^{\infty} a_{n}\left(x-x_{0}\right)^{n}, \quad\left|x-x_{0}\right|

Short answer

Question: Prove that if \(f(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^n\) with the convergence radius \(R>0\), then an antiderivative of \(f(x)\) is given by \(F(x) = C + \sum_{n=0}^{\infty} \frac{a_n}{n+1}(x-x_0)^{n+1}\) with the same convergence radius, where \(C\) is the constant of integration. Solution: We have proven that the given form of \(F(x)=C + \sum_{n=0}^{\infty} \frac{a_n}{n+1}(x-x_0)^{n+1}\) is indeed an antiderivative of the function \(f(x)= \sum_{n=0}^{\infty} a_n (x-x_0)^n\), as it satisfies \(F'(x) = f(x)\) for all values of \(x\) in the interval \(\left|x-x_0\right|R. This also demonstrates that the convergence radius of the antiderivative is the same as that of the original function.

Step-by-step solution

Start with the given power series for \(f(x)\)

Write down the given power series representation of the function \(f(x)\): $$ f(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^n, \quad \left|x-x_0\right|<R. $$

Differentiate term-by-term

Differentiate each term of the power series individually with respect to \(x\), and rearrange the terms to get a new series. Using the power rule of differentiation, $$ \frac{d}{dx}\left[a_n (x-x_0)^n\right] = na_n (x-x_0)^{n-1}. $$ So, the derivative of the entire power series is $$ f'(x) = \sum_{n=0}^{\infty} na_n (x-x_0)^{n-1}. $$

Verify the antiderivative

Now, consider the proposed antiderivative of \(f(x)\): $$ F(x) = C + \sum_{n=0}^{\infty} \frac{a_n}{n+1}(x-x_0)^{n+1}. $$ Differentiate \(F(x)\) with respect to \(x\). Since \(C\) is a constant, its derivative is zero. Using the power rule of differentiation, the derivative of the power series is $$ F'(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^n, $$ which is exactly the given function \(f(x)\). Thus, \(F'(x) = f(x)\).

Discuss the constant of integration

Since \(F'(x) = f(x)\) for all values of \(x\) in the given interval, we can conclude that \(F(x) = C + \sum_{n=0}^{\infty} \frac{a_n}{n+1}(x-x_0)^{n+1}\) is indeed an antiderivative of \(f(x)\). Here, \(C\) is the integration constant, which can be any real number. So, we have proven the result: $$ F(x) = C + \sum_{n=0}^{\infty} \frac{a_n}{n+1}(x-x_0)^{n+1}, \quad \left|x-x_0\right|<R. $$

Key concepts

Term-by-term Differentiation

To understand term-by-term differentiation, consider a power series—basically a sum of terms of the form \(a_n(x-x_0)^n\). Differentiating term-by-term simply means applying differentiation rules to each term of the series independently.
When working with power series, this differentiation is legitimate within the interval of convergence, \(\left|x-x_0\right|Term-by-term differentiation is often used because it makes the problem more tractable. The intuition behind it is that if each piece of the series behaves nicely (that is, differentiable), so should the whole. In the context of real analysis, this is a powerful tool and forms the basis of more general theorems.

Power Series Representation

A power series is essentially a function represented as an infinite sum of terms in the form \(a_n(x-x_0)^n\), where \(a_n\) are coefficients and \(x_0\) is the center of the series. It is a way of expressing functions that may be too complex or not possible to write in a closed form.

• It allows us to perform operations like differentiation and integration on the function term by term, often simplifying the process.
• It is used in various fields such as physics, engineering, and economics to solve problems involving differential equations and model complex systems.
• Each function has a radius of convergence, \(R\), within which the power series converges to the function.

The power series representation is one of the cornerstones of calculus and real analysis, providing a versatile approach to explore the behavior of mathematical functions.

Integration Constant

The integration constant \(C\) represents an indefinite integral's arbitrariness. When you integrate a function, you are essentially finding all functions whose derivative is the original function. This infinite set of possibilities results in \(+ C\), where \(C\) can be any real number.
This constant is crucial in solving initial value problems where you have additional information about the function's value at a certain point. By knowing this, you can pinpoint the exact antiderivative needed among the infinite possibilities.
Understanding how the integration constant plays into antiderivatives is essential in both calculus and real analysis, influencing the way mathematicians interpret the behavior of functions over an interval.

Real Analysis

Real analysis is a branch of mathematics dealing with real numbers and the analytic properties of real functions and sequences. It rigorously formulates and proves the theorems of calculus, providing a framework for understanding concepts like convergence, continuity, and differentiability.

• In the realm of real analysis, functions are often studied through their power series expansions.
• The field delineates conditions under which operations like term-by-term differentiation are valid.
• It provides the tools to analyze the integrities, such as the integration constant, ensuring a smooth formulation of the antiderivatives.

The principles of real analysis are not just academic; they form the backbone of numerical methods and are used to solve real-world problems in various scientific and engineering disciplines.