I am reading an example in i m sorry the writer is detect the power series representation the $\ln(1+x)$. Below is the components related to the question:

I think that ns get every little thing except because that one thing: Why do we need to uncover a specific constant $C$ and also not simply leave at together an arbitrarily constant? and why perform we discover the specific continuous we require by setup x=0 and also solve the provided equation?

$\begingroup$ The logarithm is a function, an interpretation that it has a well defined value because that a given $x$. Friend can't leaving an undetermined continuous in the meaning ! $\endgroup$
Because the is not true that us have$$\log(1+x)=x-\fracx^22+\fracx^33-\fracx^44+\cdots+C$$for an arbitrary continuous $C$. Since, once $x=0$, the LHS is $0$ and also RHS is $C$, $C=0$.

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

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