Can n be negative? What values of n allow for negative numbers?

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SUMMARY

The discussion centers on the factorial function, specifically addressing whether negative values of n can exist. Traditional factorials, defined only for natural numbers, cannot be negative. However, the Gamma function, an extension of the factorial, allows for negative inputs, producing negative outputs for certain values such as -1 and -3.74768264672741260139148848269149969586163939513235551205229915.

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If yes, then for what values of n?
 
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We usually only define ##n!## for natural numbers ##n##. Then ##n!## can never be negative.

There is however a very natural extension of the factorial, this is called the Pi function (which is just a translation of the Gamma function). This can indeed become negative as you can see from the graph: http://en.wikipedia.org/wiki/Gamma_function
 
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If we extend the domain we can have negative numbers for example
-1~(-3.74768264672741260139148848269149969586163939513235551205229915)!
-1~(-3.45702473822080062303945414765117954323659790903378442096479450)!
 
THAT is what I wanted to see. Thank you very much Sir.
 

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