The exponential generating function for the sequence of factorials is x! = Γ(x+1) = [tex]\int\limits^{infinity}_0 {t}^{x}{e}^{-t} \, dt[/tex].
We are given a sequence. A sequence in mathematics is an enumerated collection of items in which repetitions are permitted and order is important. It has members, just like a set. The length of the sequence is defined by the number of elements. The given sequence consists of factorials. The product of all positive integers less than or equal to n, denoted by n!, is the factorial of a non-negative integer n. We need to find the exponential generating function for the given sequence. The sequence is 0!, 1!, 2!, …, n!. The exponential generating function for the sequence of factorials is given below.
x! = Γ(x+1) = [tex]\int\limits^{infinity}_0 {t}^{x}{e}^{-t} \, dt[/tex]
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