where $n \geq 2 $
Given that a function containing an odd number of exponentiated terms as follows
$$\left(\frac1{n}\right)^{\left(\frac1{n+1}\right)^{.^{.^{.^{\left(\frac1{n+2m+1}\right)}}}}} $$
produces a decaying value
and
that a function containing an even number of exponentiated terms as follows
$$\left(\frac1{n}\right)^{\left(\frac1{n+1}\right)^{.^{.^{.^{\left(\frac1{n+2m}\right)}}}}} $$
produces a growing value
and
neither are represented by a continuous function .....
how would I prove convergence where either
a) they converge on their own values?
or
b) they converge on the same value?
or
c) they diverge completely?
In the end the main issue of concern is whether or not the following converges?
$$\lim_{m\to\infty}\left(\frac1{n}\right)^{\left(\frac1{n+1}\right)^{.^{.^{.^{\left(\frac1{m}\right)}}}}}$$