Could someone help me with part C of this question, relating to the Gamma Function. I am aware it is something to do with convergence but I am not sure how to show this
(a) Use integration by parts to show that $Γ(x + 1) = xΓ(x)$.
(b) Hence prove that $Γ(x + 1) = x!$ when $x$ is a positive integer.
(c) The property found in (a) can be used to extend the definition of $Γ(x)$ to some negative values of $x$. Show that a value for $Γ(0)$ cannot be defined in this way. Are there other values of $x$ for which $Γ(x)$ cannot be defined in this way?