- #1
sobolev
- 10
- 0
Can any of you solve this?
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 applied n times to x e.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 applied n times to x.
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]).
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 applied n times to x e.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 applied n times to x.
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]).