Power series for derivatives


a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.


f(x) = eˣ

Verified step by step guidance

1. Start with the Taylor series expansion of the function \(f(x) = e^x\) centered at 0, which is given by the infinite sum: \[f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]
2. Differentiate the series term-by-term with respect to \(x\). Using the power rule for differentiation, the derivative of each term \(\frac{x^n}{n!}\) is: \[\frac{d}{dx} \left( \frac{x^n}{n!} \right) = \frac{n x^{n-1}}{n!} = \frac{x^{n-1}}{(n-1)!}\] Note that for \(n=0\), the term is a constant and its derivative is zero, so the summation index will start from \(n=1\) after differentiation.
3. Rewrite the differentiated series by adjusting the index to start from \(n=0\) for convenience. Let \(m = n - 1\), then: \[f'(x) = \sum_{m=0}^{\infty} \frac{x^m}{m!}\]
4. Recognize that the differentiated series is the same as the original Taylor series for \(e^x\). Therefore, the function represented by the differentiated series is also \(e^x\).
5. Determine the interval of convergence. Since the original series for \(e^x\) converges for all real \(x\), the differentiated series will have the same interval of convergence, which is \((-\infty, \infty)\).

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point, often zero (Maclaurin series). For eˣ, the series is ∑(xⁿ/n!), which converges for all real x. Understanding this expansion is essential to differentiate the series term-by-term.

Term-by-Term Differentiation of Power Series

Power series can be differentiated term-by-term within their interval of convergence. Differentiating each term of the Taylor series of eˣ involves applying the power rule to xⁿ/n!, resulting in a new series representing the derivative function. This process helps identify the derivative's power series.

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges. Differentiating a power series does not change its radius of convergence, so the interval for the derivative's series is the same as the original. For eˣ, the series converges for all real numbers, meaning the interval is (-∞, ∞).