i am reading an instance in i beg your pardon the author is detect the power series representation that $ln(1+x)$. Here is the components related come the question:

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



$egingroup$ The logarithm is a function, definition that it has a well characterized value because that a offered $x$. Girlfriend can't leaving an undetermined constant in the meaning ! $endgroup$
Because it is not true that we 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 duty is $log (1+x)$ and for $x=0$ we have $log (1+0)=0$ we need that likewise the collection is zero because that $x=0$.


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