The gamma function, which plays an important role in advanced applications, is defined for n ≥ 1 by

(a) Show that the integral defining Γ(n) converges for n ≥ 1 (it actually converges for all n > 0). Show that tⁿ⁻¹e^−t −2 for t sufficiently large.
(b) Show that Γ(n + 1) = nΓ(n) using Integration by Parts.
(c) Show that Γ(n + 1) = n! if n ≥ 1 is an integer. Use (b) repeatedly. Thus, Γ(n) provides a way of defining n-factorial when n is not an integer.

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= 5²°₁ I(n) = t-le-¹ dt