Problem 59 - Thomas Calculus

Question

Uniqueness of convergent power series a. Show that if two power series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) and \(\sum_{n=0}^{\infty} b_{n} x^{n}\) are convergent and equal for all values of \(x\) in an open interval \((-c, c),\) then \(a_{n}=b_{n}\) for every \(n\) . (Hint: Let \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}=\sum_{n=0}^{\infty} b_{n} x^{n} .\) Differentiate term by term to show that \(a_{n}\) and \(b_{n}\) both equal \(f^{(n)}(0) /(n !) . )\) b. Show that if \(\sum_{n=0}^{\infty} a_{n} x^{n}=0\) for all \(x\) in an open interval \((-c, c),\) then \(a_{n}=0\) for every \(n .\)

Step-by-Step Solution

Answer

The coefficients \(a_n\) and \(b_n\) are equal for all \(n\), and if the series is zero, then each \(a_n\) is also zero.

Step 1: Understanding the problem

We have two power series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) and \(\sum_{n=0}^{\infty} b_{n} x^{n}\) that converge and are equal on an open interval \((-c, c)\). We need to show that the coefficients \(a_n\) and \(b_n\) are equal for every \(n\). Also, if one of these series is zero for all \(x\) in \((-c, c)\), all coefficients must be zero.

Step 2: Define the function

Define \(f(x) = \sum_{n=0}^{\infty} a_{n} x^{n} = \sum_{n=0}^{\infty} b_{n} x^{n}\). Since \(f(x)\) represents both series, it follows that its derivatives will also be equal, leading us to a way to determine the coefficients.

Step 3: Differentiate term by term

The differentiation of \(f(x)\) term by term gives the derivatives \(f'(x) = \sum_{n=1}^{\infty} n a_{n} x^{n-1} = \sum_{n=1}^{\infty} n b_{n} x^{n-1}\). Continuing this process, we recognize that \(f^{(n)}(x) = n! a_{n} + O(x) = n! b_{n} + O(x)\), hence \(a_n = b_n\).

Step 4: Evaluate derivatives at zero

By evaluating the derivatives at zero, \(f^{(n)}(0) = n! a_n = n! b_n\). Therefore, \(a_n = f^{(n)}(0)/n! = b_n\), proving \(a_n = b_n\) for all \(n\). Thus, the coefficients of the power series are uniquely determined.

Step 5: Special case for zero series

When \(\sum_{n=0}^{\infty} a_{n} x^{n} = 0\) for all \(x\) in \((-c, c)\), we apply the same reasoning: the only way a derivative like \(f^{(n)}(0)\) can be zero for all \(n\) is if each \(a_n = 0\). This ensures the equality holds in the open interval.

Key Concepts

Convergent Power Series

A power series is a series of the form \(\sum_{n=0}^{\infty} a_{n} x^{n}\), where \(a_n\) represents the coefficients and \(x\) is the variable. A convergent power series is one where, for each value of \(x\) within a certain range called the interval of convergence, the series approaches a finite value. In the problem at hand, we are dealing with two convergent power series that are equal for every \(x\) in an open interval \((-c, c)\).
This convergence is crucial because it allows us to differentiate the power series term by term and use analytic methods to explore the series' behavior and properties. When you differentiate a power series term by term within its radius of convergence, it ensures that each derivative has its own power series, which is still convergent within the same interval. These properties will help us prove that these series are not just equal overall, but equal term by term, leading to each of their coefficients being exactly the same.
This convergence also plays a key role in the concept of function equality within the interval, as equal functionality implies identical behaviors and properties, such as equal derivatives which point directly to equal coefficients in the power series.

Equal Coefficients

The idea that two convergent power series having equal coefficients stems from their differential properties. If \(f(x) = \sum_{n=0}^{\infty} a_{n} x^{n}\) and \(g(x) = \sum_{n=0}^{\infty} b_{n} x^{n}\) are equal for all \(x\) in \((-c,c)\), their coefficients \(a_n\) and \(b_n\) must be equal. This can be shown by considering the uniqueness of derivative expressions.
By differentiating both series term by term, we get that \(f^{(n)}(x) = n! a_n x^{n-n} + \text{higher terms}\) and \(g^{(n)}(x) = n! b_n x^{n-n} + \text{higher terms}\). Because the series are equal, their derivatives at \(x=0\) must also be equal, so \(n! a_n = n! b_n\). This implies that \(a_n = b_n\) for all \(n\).
This process highlights the importance of differentiability and the function's ability to be expressed as a power series to conclude that equal functions in some interval at all points are determined by the equality of their coefficients. It explains how power series possess this unique quality where the entire function is defined uniquely by its coefficients, assuming convergence and equality within an interval.

Infinite Series

An infinite series is a sum of infinitely many terms, represented generally as \(\sum_{n=0}^{\infty} a_{n} x^{n}\). In the realm of mathematics, infinite series allow us to describe functions that might not be easily expressible otherwise and often reveal deeper properties of these functions.
Considering the problem, when a power series is zero throughout an entire interval \((-c, c)\), each term within this series must contribute nothing to the sum for all values of \(x\). Therefore, these conditions force every individual coefficient \(a_n\) to be zero, proving that \(a_n = 0\) for every \(n\). This showcases the idea that a function represented by an infinite series is fully defined by its coefficients, and if all coefficients are zero, then the function itself is zero.
Infinite series come up frequently in calculus when discussing convergence and serve as fundamental tools in understanding the behavior of polynomials and transcendental functions over an interval. The phenomenon where \(\sum_{n=0}^{\infty} a_{n} x^{n} = 0\) requires each \(a_n\) to also be zero ensures the integrity of power series as representing well-defined functions, emphasizing their uniqueness and precision.