The Power Series definition for tetration

[Gottfried]

What would be the Power Series for X tetrated to the 2nd, X tetrated to the 3rd, 4th, ...etc. Then find a pattern in these series for a general series definition of X Tetrated to any X.

The Series for:
X tet 2 = X ^ X = exp ( X ln X )=

Sigma ((X ln X) ^ n) / n!
for n = 0 to infinity.
now let this series = A

Then
the series for X tet 3 = X ^ A = exp (A ln X) =

Sigma ((A ln X) ^ m)) / m!
for m = 0 to infinity
now let this series = B

Then continue the process for more higher nested series...

Can anyone express the nested series for the tetration powers of 2 and higher as just a single power series?

IF this can be shown, is their any pattern to these Sigma expressions to give a generalized power series ?

can X tet X be decribed using product series?

Hmm, what prevents you, to just to try this, and show us what you get for tet 2, tet 3... ?
For decremented exponentiation you may find this article interesting.
powerseries iteration