What is the relation/connection between $n!$ or $e$ and $2^n$ ? Is the there a relation/connection between $n!$ or $e$ and $2^n$?

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    $\begingroup$ Define 'relation.' There's Stirling's approximation that gives an asymptotic formula for $n!$ involving $e^n$. $\endgroup$ – user61527 Mar 9 '14 at 16:37

The first thing in my head after reading the question:$$\sum_{n=0}^{\infty} \dfrac{2^n}{n!} = e^2$$

  • $\begingroup$ Very nice! I've thought a bit towards this and that - but your proposal is so simple :+) $\endgroup$ – Gottfried Helms Mar 9 '14 at 16:46
  • $\begingroup$ what is the intuition behind that $e$ takes the other's bottom on its shoulder, in summations? $\endgroup$ – user132079 Mar 9 '14 at 16:55
  • $\begingroup$ There was no intuitive reasoning, simply a definition. As a power series, we have (and can define) $e^x = \displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{n!}$, which converges for all complex numbers. You happened to ask about functions/numbers that would appear in the case $x=2$(!) $\endgroup$ – FireGarden Mar 9 '14 at 17:01
  • $\begingroup$ thats tautological, perhaps my problem is with *+-% etc. Anyway:) $\endgroup$ – user132079 Mar 9 '14 at 17:17
  • $\begingroup$ in response to @waqarahmad comment ''what is the intuition behind that e takes the other's bottom on its shoulder, in summations?'' Formal Power Series $\endgroup$ – waqar ahmad Mar 24 '14 at 8:02

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