Don't know why this power series representation is wrong...

Question

I've run into something confusing.

The problem is that I have to find the power series representation of $g(x)$ using the given $f(x)$, specifically $g(x) = \ln(1 - 3x)$ using $f(x) = \frac{1}{1-x}$.

Now, I did it, but my result is incorrect: $$\ln(1-3x) = \int \frac{dx}{1-3x} = \int \sum 3^{k}x^{k} = \sum \frac{3^{k}x^{k+1}}{k+1}$$

The answer at the back of my textbook is: $$\sum \frac{3^k x^k}{k}$$

My result for the interval of convergence was however correct, which leads me to believe that there is maybe a mistake in the book's answer?

Any help is appreciated, thanks!

Answer 1

You're simply neglecting the chain rule: $$ \int\frac{dx}{1-3x} = \frac {-1} 3 \ln(1-3x) + C. $$

Answer 2

Note that $$\ln(1-3x) = -3\int \dfrac{dx}{1-3x}$$ and then shift the index of summation.

Answer 3

Use the chain rule: $$\ln(1-3x)=\int\dfrac{dx}{1-3x}$$ Let $u=1-3x,du=-3dx$, thus giving $$\ln(1-3x)=\dfrac{-1}{3}\int\dfrac{du}{u}=\dfrac{-1}{3}\ln|1-3x|+C$$ Thus, $$\ln(1-3x)=\int\dfrac{dx}{1-3x}=\sum\dfrac{3^kx^k}{k}$$