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Exponential formula may be defined: $$f(x)=\sum_{i=0}^\infty x^n/n!$$

By multiplying together the two series expansions for f(x) and f(y) and collecting the first few terms of the same degree (e.g. $x^3$, $x^2y$,$xy^2$ and $y^3$ are all of degree 3), verify that

(1) $f(x)f(y)=f(x+y)$

Show that the terms of degree n in $f(x)f(y)$ and $f(x+y)$ are the same. Does this prove (1)?

Hi guys I have no idea how to solve this. Could you help me, perhaps?

Simply Beautiful Art
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user491897
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  • I know it seems a daunting task, but all you really have to do is follow the question lead: collect the terms $x^ay^b $ for $a+b $ constant. Have you tried that? – RGS Oct 15 '17 at 22:11
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    See this answer. – Simply Beautiful Art Oct 15 '17 at 22:15
  • Since they already tell you that $f(x)\cdot f(y)=f(x+y)$ (well, yeah $e^x\cdot e^y=e^{x+y}$), you could also start from looking at $f(x+y)$ and use the binomial expansion (http://mathworld.wolfram.com/BinomialTheorem.html) – user12345 Oct 15 '17 at 23:07
  • Someone should write this up as an answer, so it can be accepted, or the op should delete this question, because this question has be answer, but it is marked as unanswered. – ViktorStein Nov 06 '18 at 07:49

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