Consider the famous Gamma function, defined by Γ(x) = R [infinity] 0 t x−1 e −t dt. Using mathematical induction, prove that Γ(n + 1) = n! for any n ∈ N ∪ {0}, where n! = n(n − 1)· · · 3 · 2 · 1 is the usual factorial function

See proof below

Step-by-step explanation:

The inductive proof consists on the following steps:

1) Base case: for n=0, we will prove that Γ(1)=0!=1. We have that

Hence the base case holds.

2) Inductive step: suppose that Γ(n + 1) = n! for some natural number n. We will prove that Γ((n + 1)+1) = (n+1)!

Use integration by parts, with the following parts:

u=t^{n+1}, du=(n+1)t^n

dv=e^{-t}, v=-e^{-t}

and we used the induction hypotheses on this last line. Also, -t^n e^-t tends to zero as n tends to infiity (the exponential decays faster than any polynomial).

We have proved the statement for n+1, and by mathematical induction, the statement holds for all n.

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