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) = ln (1 + x)

Verified step by step guidance

1. Start with the Taylor series expansion of the function \(f(x) = \ln(1 + x)\) centered at 0. Recall that the Taylor series for \(\ln(1 + x)\) is given by the alternating series \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}\) for \(|x| < 1\).
2. Differentiate the series term-by-term with respect to \(x\). Use the rule \(\frac{d}{dx} \left( x^n \right) = n x^{n-1}\). Applying this to each term, the differentiated series becomes \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{d}{dx} \left( \frac{x^n}{n} \right) = \sum_{n=1}^{\infty} (-1)^{n+1} x^{n-1}\).
3. Rewrite the differentiated series to identify the function it represents. Notice that \(\sum_{n=1}^{\infty} (-1)^{n+1} x^{n-1}\) can be re-indexed or recognized as a geometric series with alternating signs. This series corresponds to \(\frac{1}{1+x}\) for \(|x| < 1\).
4. Confirm the function represented by the differentiated series is indeed \(f'(x) = \frac{1}{1+x}\), which matches the derivative of \(f(x) = \ln(1+x)\).
5. Determine the interval of convergence for the differentiated series. Since the original series converges for \(|x| < 1\) and differentiation does not change the radius of convergence, the interval of convergence remains \((-1, 1)\). Check endpoints separately if needed.

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). It allows complex functions to be expressed as power series, facilitating differentiation and integration 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 individually produces a new power series representing the derivative of the original function, preserving convergence properties inside the interval.

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, but the behavior at endpoints must be checked separately to determine the exact interval for the derivative series.