Solution: Combining power series Use the power series

Chapter 8, Problem 30E

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QUESTION:

Combining power series Use the power series representation

\(f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k), for \(-1 \leq x<1\),

to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.

\(f\left(x^{3}\right)=\ln \left(1-x^{3}\right)\)

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QUESTION:

Combining power series Use the power series representation

\(f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k), for \(-1 \leq x<1\),

to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.

\(f\left(x^{3}\right)=\ln \left(1-x^{3}\right)\)

ANSWER:

Solution 30E
Step 1:

Given
        Using the power series representation for
, we get
        

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