i am reading an instance in i m sorry the writer is finding the power series representation the $ln(1+x)$. Here is the components related to the question:

I think that i get everything except because that one thing: Why execute we require to discover a specific continuous $C$ and also not simply leave at as an arbitrarily constant? and also why perform we uncover the specific consistent we need by setting x=0 and solve the given equation?



$egingroup$ The logarithm is a function, definition that it has a well defined value for a offered $x$. Girlfriend can't leaving an undetermined constant in the an interpretation ! $endgroup$
Because it is not true that we have$$log(1+x)=x-fracx^22+fracx^33-fracx^44+cdots+C$$for an arbitrary constant $C$. Since, when $x=0$, the LHS is $0$ and RHS is $C$, $C=0$.

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Since the original duty is $log (1+x)$ and for $x=0$ we have actually $log (1+0)=0$ we need that likewise the series is zero for $x=0$.


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