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I came across this power series while solving a problem.

$$1-{x\over 2}+{x^2\over 3}-{x^3\over4}+\dots$$

I calculated its radius of convergence and it turned out to be $1$.

Does this power series represent some well function in the given interval of convergence?

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For $1\ge x>-1,$

$$\ln(1+x)=x-\dfrac{x^2}2+\dfrac{x^3}3-\dfrac{x^3}4+\cdots$$

Divide both sides by $x$ assuming $x\ne0$

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