Question

Power series for derivativesa. 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

Accepted Answer

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 $$ |x| \u003c 1 . $$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 $$ \sum_{n=1}^{\infty} n x^{n-1} . $$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. Determine the interval of convergence for the differentiated series. Since the original series converges for $$ |x| \u003c 1 $$, and differentiation does not change the radius of convergence, the interval remains $$ (-1, 1) . $$Summarize: The differentiated power series is $$ \sum_{n=1}^{\infty} n x^{n-1} $$, representing the function $$ (1 - x)^{-2} $$, and it converges for $$ |x| \u003c 1 .$$

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

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Concetti chiave

Taylor Series and Power Series

Term-by-Term Differentiation of Power Series

Interval of Convergence