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I am looking at examples of Maclaurin expansions for different functions, such as e^x, and sinx. But there is no expansion for log(x), only log(x+1). Why is that?
So why does it have log(x+1), and not log(x+1/2) or log(x+2), for example?The Maclaurin series uses the values of the function and it's derivatives at x=0. But log(x) is not defined at x=0.
Mostly convenience, I suppose. Each of the other two expressions could also be expanded as Maclaurin series.So why does it have log(x+1), and not log(x+1/2) or log(x+2), for example?
Expanding the Maclaurin series at x=0 would be trying to evaluate the log at negative numbers.Log((x+1)/(x-1)) gives a series that can be used for any y=(x+1)/(x-1)