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) = ln (1 + x)

Accepted Answer

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 $$|x| \u003c 1. $$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} $$(-1)^{n+1}$$ $$x^{n-1}. $$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| \u003c 1$$. Confirm the function represented by the differentiated series is indeed f$$'(x) = \frac{1}{1+x}$$, which matches the derivative of f(x$$) = \ln(1+x)$$. Determine the interval of convergence for the differentiated series. Since the original series converges for $$|x| \u003c 1$$ and differentiation does not change the radius of convergence, the interval of convergence remains (-1$$, 1)$$. Check endpoints separately if needed.

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)

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Key Concepts

Taylor Series Expansion

Term-by-Term Differentiation of Power Series

Interval of Convergence