Problem 59 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

If power series are equal on an interval, coefficients must be equal. If a series equals zero, all coefficients are zero.

Step 1: Set Up the Problem for Part A

We are given 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 for all \( x \) in an open interval \((-c, c)\). Define a function \( f(x) = \sum_{n=0}^{\infty} a_{n} x^{n} = \sum_{n=0}^{\infty} b_{n} x^{n} \). Our goal is to show \( a_n = b_n \) for every \( n \).

Step 2: Differentiating the Function

Differentiate \( f(x) \) term by term. The derivative is given by \( f'(x) = \sum_{n=1}^{\infty} n a_{n} x^{n-1} = \sum_{n=1}^{\infty} n b_{n} x^{n-1} \). Evaluating at \( x = 0 \), we have \( f'(0) = a_1 = b_1 \).

Step 3: Generalize with Higher Derivatives

Compute higher order derivatives of \( f(x) \). Compute \( f^{(n)}(x) = \sum_{n=n}^{\infty} n(n-1)...(n-(m-1)) a_{n} x^{n-m} \) and evaluate at \( x = 0 \). This gives \( f^{(n)}(0) = n! a_n = n! b_n \). Therefore, \( a_n = b_n = f^{(n)}(0)/n! \).

Step 4: Applying the Result to Part B

In part b, assume \( \sum_{n=0}^{\infty} a_{n} x^{n} = 0 \) for all \( x \) in \((-c, c)\). Then the function \( f(x) = \sum_{n=0}^{\infty} a_{n} x^{n} = 0 \). This implies that all derivatives \( f^{(n)}(0) = 0 \). For \( n = 0,...,\infty \), this results in \( a_n = f^{(n)}(0)/n! = 0 \).

Step 5: Conclusion for Part B

Since for each \( n \), \( a_n = 0 \), it follows that \( a_n = 0 \) for every \( n \) when the series \( \sum_{n=0}^{\infty} a_{n} x^{n} \) converges to zero throughout the interval \((-c, c)\).

Key Concepts

Term by Term Differentiation

When dealing with power series, "term by term differentiation" allows us to differentiate the entire series by differentiating each term individually. This technique is especially useful when considering power series functions that converge on a certain interval.
Let's consider a power series expressed as \( f(x) = \sum_{n=0}^{\infty} a_n x^n \). When this series is differentiable within an open interval around zero, we can find its derivative by differentiating each term of the series:

• The derivative of \( a_n x^n \) is \( n a_n x^{n-1} \).
• Thus, the term by term differentiated series becomes \( f'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1} \).

This method simplifies the process of finding derivatives of power series functions without needing to rewrite the series in a different form or lose details at each term. Evaluating this derivative at \( x = 0 \) in the given example helps determine coefficients such as \( a_1 \) since \( f'(0) = a_1 = b_1 \), understanding the crucial role derivatives play in analyzing power series.

Higher Order Derivatives

Higher order derivatives of a power series are a straightforward extension of basic derivatives. These derivatives help us access more in-depth behaviors and properties of the series. Calculating these derivatives involves repeated term by term differentiation.
For instance, if the function \( f(x) = \sum_{n=0}^{\infty} a_n x^n \), then the second derivative is obtained by differentiating \( f'(x) \) term by term:

• The second derivative is \( f''(x) = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} \).

Subsequent derivatives continue this pattern. Evaluating the n-th derivative at \( x = 0 \) gives us \( f^{(n)}(0) = n! a_n \), illustrating how the nth coefficient \( a_n \) can be expressed in terms of this derivative. This pattern holds for each derivative until we can conclude that if two power series are equal on a certain interval, their coefficients must also be equal at every term. The fact that each derivative at zero defines exactly one coefficient is vital in ensuring the uniqueness of series representation.

Convergence in Open Interval

Convergence of a power series refers to the set of points \( x \) for which the series sums to a finite value. An "open interval" is a set of real numbers between two endpoints where the endpoints are not included. Power series often converge within such intervals.When a power series like \( f(x) = \sum_{n=0}^{\infty} a_n x^n \) converges in an open interval \((-c, c)\), it transforms into a useful function within this range.
If two different power series converge within an open interval and are known to be equal, their coefficients must necessarily be identical. By differentiating the series term by term, we determine these coefficients, allowing us to prove their uniqueness, as shown in the solution. Similarly, if a power series sums to zero over an entire open interval, then each coefficient must be zero, leading to the conclusion that the function is identically zero throughout that interval. This aspect of convergence is critical in function analysis and shows how conditions within an interval can determine properties of the series.