Check the associativity of formal series

Asked May 18 '23 at 05:44

Active May 18 '23 at 05:44

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I am studying for an exam and I am stuck on solving this problem:

Check the associativity for the formal series $F(G(H(z))) = F(G(z))(H(z))$ whenever it is defined.

In one hand, it seems straightforward and intuitive to me, but at the same time I can't find a way how to prove the associativity. But it has to be something simple right?

asked May 18 '23 at 05:44

user180523

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Please explain your notation – azif00 May 18 '23 at 05:49
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For example, if $F(z)$,$G(z)$ are formal power series, then $F(G(z))$ is often not defined. Perhaps you would want $G(0)=0$. Then the coefficients of $F(G(z))$ are only finite sums involving the coefficients of $F$ and $G$. And we can get any given term of $F(G(z))$ using only truncations of $F,G$. Perhaps this will reduce the question to the case of polynomials. – GEdgar May 18 '23 at 06:22
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Does this answer your question? Composition of Formal Power Series is associative – Anne Bauval May 18 '23 at 06:30

Check the associativity of formal series

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