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) = (1 − x)⁻¹

Verified step by step guidance

1. Start with the given function \( f(x) = (1 - x)^{-1} \). Recall that its Taylor series centered at 0 (Maclaurin series) is the geometric series \( \sum_{n=0}^{\infty} x^n \), valid for \( |x| < 1 \).
2. Differentiate the series term-by-term. The derivative of \( f(x) \) is \( f'(x) = \frac{d}{dx} (1 - x)^{-1} \). Using the power series, differentiate each term \( x^n \) to get \( n x^{n-1} \). So the differentiated series is \( \sum_{n=1}^{\infty} n x^{n-1} \).
3. Identify the function represented by the differentiated series. Differentiate \( f(x) = (1 - x)^{-1} \) directly using the chain rule: \( f'(x) = (1 - x)^{-2} \). This matches the sum of the differentiated series, confirming the function represented by the series.
4. 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 remains \( (-1, 1) \).
5. Summarize: The differentiated power series is \( \sum_{n=1}^{\infty} n x^{n-1} \), representing the function \( (1 - x)^{-2} \), and it converges for \( |x| < 1 \).

Key Concepts

Taylor Series and Power Series

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 expresses functions as power series, allowing approximation and analysis of functions using polynomials.

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, so the interval for the derivative series is the same or possibly smaller, requiring verification at endpoints.