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Firstly, some notation:

Let [tex]\Pi(x) = \Gamma(x+1)[/tex] where [tex]\Gamma(x)[/tex] is the usual gamma function i.e. an extension of the factorial to the complex numbers.

Let [tex]log^{n} (x) = log( log( \cdots log( x ) ) )[/tex] where [tex]log[/tex] is appliedntimes toxe.g. [tex]log^{4} (x) = log( log( log( log( x ) ) ) )[/tex].

Similarly, let [tex]{\Pi}^{n} (x) = \Pi( \Pi( \cdots \Pi( x ) ) )[/tex] where [tex]\Pi[/tex] is appliedntimes tox.

THE QUESTION:

Let [tex]a_{n} = log^{n} ( {\Pi}^{n} (3) )[/tex].

Evaluate, using a computer of otherwise, [tex]\lim_{n \rightarrow \infty}{ a_{n} }[/tex] to five decimal places.

(If you are feeling especially clever try to derive a closed form expression for [tex]\lim_{n \rightarrow \infty}{ a_{n} }[/tex]).

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# Hard Problem

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