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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