How to express the infinite power tower $a^{{{{{{(a+1)}^{(a+2)}}^{(a+3)}}^{.}}^{.}}^{.}}$?

Question

Just a relatively simple question; I'm just wondering what would be the proper notation to use to express an infinite power tower that has each repeated exponent increasing by a value of $1$, like such;

$$a^{{{{{{(a+1)}^{(a+2)}}^{(a+3)}}^{.}}^{.}}^{.}}$$

Answer 1

for $0\leq a<1$ the expression become $0$. for $a=1$ it's $1$. For the rest it is $+\infty$ or $-\infty$
But, I read somewhere $n¡=n^{{n-1}^{{{n-2}^.}^.}}.$

note: $n¡$ is the factorial notation $!$ turned upside down. $¡$ keeps exponentiating while $!$ keeps multiplying. Here is an example of it.
It seems you are interested in infinite Power Tower $($Tetration with infinite height$)$. For further information visit here and wikipedia also.

Answer 2

When $a>1,$ $(a+1)^{(a+2)^{(a+3)^{\cdots}}}\ge 2^{3^{4^{\cdots}}}$ will obviosuly go to infinity, and thus $a^{(a+1)^{(a+2)^{(a+3)^{\cdots}}}}$ will also go to infinity.

When $a=1,$ the power tower will clearly go to $1.$

When $0<a<1,$ $(a+1)^{(a+2)^{(a+3)^{\cdots}}}$ will go to infinity, as seen above. Thus, $a^{(a+1)^{(a+2)^{(a+3)^{\cdots}}}}$ will go to $0.$

When $a=0,$ the power tower will clearly go to $0.$