I"m doing a sequences problem where I have to write the an initial five terms of a sequence. The looks normal, however there is an exclamation note on the denominator:

$$a_n = frac1(n + 1)!$$

&

$$a_n = frac(-1)^nnn! + 1$$

What walk the exclamation note mean, and how execute I go about this differently with the exclamation mark?




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The factorial that a number is represented by the exclamation allude (!).The factorial of a number $x$ is often described as the product that all positive integers much less then or same to $x$. Because that example:

$$4! = 4cdot3cdot2cdot1 = 24$$

It is often also useful to define $x!$ in a recursive relation:

$$x! = x(x-1)!$$

where $0! = 1$.

This method is often an excellent because the helps explain why $0!=1$ (see also the "empty product" for an ext on this). Using the over explanation, us may uncover $4!$:

$$4! = 4(3)! = 4(3)(2)! = cdots = 4(3)(2)(1)(0)! = 4(3)(2)(1)(1) = 24$$


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edited Apr 9 "12 at 1:29
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Michael continuous
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answer Apr 9 "12 at 0:09
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If the number is 4! climate you count under so it would certainly be $4 imes 3 imes 2 imes 1= 24$. An additional thing is that if the number is 7! you would count down until you acquire to two. So it would certainly be $7 imes 6 imes 5 imes 3 imes 2 =5040$ 7 x 6 x 5 x 4 x 3 x 2 I just put in the 4 the was not provided although the price is quiet 5040.


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edited Apr 27 "16 at 2:46
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CommunityBot
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answered might 19 "14 in ~ 23:03
HockeyloverHockeylover
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Hockeylover It's actually defined recursively and also one "counts down" to $0! :=1$. $endgroup$
–user137731
january 7 "15 in ~ 20:55


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