I've just come across formal power series and am not very fluent with them yet.

I'd like to show that $\exp(\sum_{n=1}^\infty a_nX^n)=\prod_{n=1}^\infty\exp(a_nX^n)$.

Can anybody help?

  • $\begingroup$ This identity holds modulo $X^N$ for all $N$. $\endgroup$ – Siméon Aug 29 '14 at 15:05
  • $\begingroup$ @Simeon Oh of course! I feel so stupid. Thanks! $\endgroup$ – Matt Gallagher Aug 29 '14 at 16:38

$$\exp\left(\sum\limits_{n=0}^{+\infty}a_n X^n\right)=\exp\left({\lim\limits_{n\to +\infty}\sum\limits_{k=0}^{n}a_k X^k}\right)=\lim\limits_{n \to +\infty}\exp \left(\sum\limits_{k=0}^{n}a_k X^k\right)=\lim\limits_{n \to \infty}\prod\limits_{k=0}^{n}\exp \left(a_k X^k\right)=\\=\prod\limits_{n=0}^{+\infty}\exp \left(a_n X^n\right) $$ We changed $\lim$ and $\exp$ because of the fact that $\exp x$ is continuous

  • $\begingroup$ Okay great thanks. Do you have a proof or reference that exp is continuous in the formal power series ring? $\endgroup$ – Matt Gallagher Aug 27 '14 at 11:28
  • $\begingroup$ As I know, $|\exp x -\sum\limits_{k=0}^{n}\frac{x^k}{k!}|<\varepsilon$, and usually one considers $\sum\limits_{n=0}^{+\infty}a_n x^n$ as a continuous function where the series converges, and the series for exp converges everywhere $\endgroup$ – cool Aug 27 '14 at 11:41
  • $\begingroup$ You'd better have $a_0=0$ for this to make sense, even with formal power series. $\endgroup$ – Ted Shifrin Aug 27 '14 at 11:47
  • $\begingroup$ @Paul I'm considering $\exp$ as a function on the formal power series ring. So I'm considering $\sum_{n=0}^\infty a_nX^n$ as a variable, not a function itself. I think your answer above runs through fine so long as $\exp$ is continuous. I imagine it is! $\endgroup$ – Matt Gallagher Aug 27 '14 at 11:50
  • $\begingroup$ Thanks @Ted, do you know if there's a way to make it work when $a_0 \neq 0$? In particular I'm interested in the case $a_0=1$. $\endgroup$ – Matt Gallagher Aug 27 '14 at 11:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.