I was recently considering a progression where each term in the sequence is the previous term raised to a common exponent.
To elucidate:
$$S_{E.P}(a,m)=a,a^m,{(a^m)}^m,({(a^m)}^m)^m \cdot \cdot \cdot $$ $$=>S_{E.P}(a,m)=a^{m^0},a^{m^1},a^{m^2},a^{m^3} \cdot \cdot \cdot $$
Discernably, the $n^{th}$ term in the sequence is $a^{m^{n-1}}$.
While arriving at a general formula for the product of terms of this sequences is very easy, I'm having difficulty arriving at a generalized formula for the sum of terms of a particular sequence. Is there any concrete way for me to attempt this in a clever manner? Does a generalized formula even exist for such a sequence?
As far as I know there is no closed formula. Evaluation by the Euler-Maclaurin formula is also out of reach.
Anyway, the exponents are increasing exponentially so that in many cases the last term is largely dominant ($|a|>1$ and the series diverges) or quickly neglectible ($|a|<1$).