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Good evening!

We all know that $\exp(x) = \sum\limits_{k=0}^\infty \frac{x^k}{k!}$ holds for all $x\in\mathbb{R}$ and $\log(x) = \sum\limits_{j=1}^\infty (-1)^{j-1}\frac{(x-1)^j}{j}$ holds for $|x-1|<1$. Further, by definition, we have $$\exp\circ\log = \operatorname{id} \text{ }(*).$$

So, a "natural" question nobody asks seems to be: How can one prove $(*)$ by using the above Taylor expansions? Even though little to nothing could be gained from it, I am curious to see if someone can "elementarily" (i.e. without using further properties of $\exp$ and $\log$) prove $$\sum\limits_{k=0}^\infty \frac{ \left(\sum\limits_{j=1}^\infty (-1)^{j-1}\frac{(x-1)^j}{j} \right)^k}{k!}=x$$ for $x\in\mathbb{R}$ with $|x-1|<1$.

Symplectic Witch
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    Your formula for log is not for $|x|<1$ but for $|x-1|<1.$ – Anne Bauval Oct 17 '22 at 22:37
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    The power series for $\log x$ is for $|x-1|<1.$ $\log x$ isn't even defined for all $|x|<1.$ – Thomas Andrews Oct 17 '22 at 22:39
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    Better to try to show $\exp(\log(1+x))=1+x$ for $|x|<1.$ – Thomas Andrews Oct 17 '22 at 22:41
  • @ThomasAndrews You're right, that would certainly simplify things. – Symplectic Witch Oct 17 '22 at 22:42
  • @TheoBendit Thanks for the link! – Symplectic Witch Oct 17 '22 at 22:43
  • It's not true that "no one seems to be asking" this question, in fact variations of this question have been asked several times on math.SE. It's also not true that "little to nothing could be gained from it"; some of the proofs readily generalize to $x$ in a Banach algebra, say, while others tell you the combinatorial content of this identity (it is equivalent to the existence and uniqueness of cycle decomposition for permutations) which generalizes to many much less obvious identities, e.g. a generating function for the number of involutions, all sorts of stuff. – Qiaochu Yuan Oct 17 '22 at 22:56
  • Here are two previous threads where I explain how to prove this using the existence and uniqueness theorems for ODEs: https://math.stackexchange.com/questions/3852168/proving-that-the-exponential-and-logarithm-functions-for-matrices-are-inverses-o/3852183#3852183, https://math.stackexchange.com/questions/2486706/how-do-i-show-that-formal-logarithm-is-the-inverse-of-the-formal-exponential/2486731#2486731 – Qiaochu Yuan Oct 17 '22 at 22:56

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