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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