I have the following power series: $x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\cdots$.
I want to use the sum of the geometric series $\sum^\infty_\limits{k=0}{x^k}=\frac{1}{1-x}$ to write a function $f(x)$ for the power series above.
When I look at the series I see that its equal to $\sum^\infty_\limits{k=0}{\frac{x^{k+1}}{k+1}}$, which is the integral of the geometric power series.
$$\int{\sum^\infty_{k=0}{x^k}=\int\frac{1}{1-x}}\implies \sum^\infty_{k=0}{\frac{x^{k+1}}{k+1}}=-\ln|1-x|$$
The function expression should be $f(x)=-\ln|1-x|$.
Is it correct to do it this way? Am I missing constant values after integrating?