Question

What is 0.5! And how do you find it (can it be found without calculus)?

Answer 1

`0.5! = sqrt(pi)/2`

Explanation:

`"Decimal factorials are defined by the Gamma function : "`

`Gamma(t) = int_0^oo x^(t-1) e^-x dx`

`"So "Gamma(t+1) = t!`

`"So to calculate 0.5! one must solve the following definite integral"`

`0.5! = int_0^oo sqrt(x) e^-x dx`

`"There exist approximating series, but that is also calculus, so"`
`"i think it cannot be solved without calculus."`

`"There exist good approximating formulas though, like Gosper's"`
`"approximation : "`

`n! ~~ (n/e)^n sqrt(pi(2n+1/3))`

`"(Gosper's approximation is an improvement to Stirling's"` `"formula)"`