Power series representation of $\ln(1+x)$?

Question

I am reading an example in which the author is finding the power series representation of $\ln(1+x)$. Here is the parts related to the question:

I think that I get everything except for one thing: Why do we need to find a specific constant $C$ and not just leave at as an arbitrary constant? And why do we find the specific constant we need by setting x=0 and solve the given equation?

Answer 1

Because it is not true that we have$$\log(1+x)=x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4+\cdots+C$$for an arbitrary constant $C$. Since, when $x=0$, the LHS is $0$ and RHS is $C$, $C=0$.

Answer 2

Since the original function is $\log (1+x)$ and for $x=0$ we have $\log (1+0)=0$ we need that also the series is zero for $x=0$.