Closed form for $\sum_{k=0}^{n} x^{a^k}$

Asked Aug 07 '16 at 22:28

Active Aug 07 '16 at 22:28

Viewed 216 times

Is there a closed form for $y=\sum_{k=0}^{n} x^{a^k}$ Where $a$ is constant?

$y=x+x^{a}+x^{a^2}+x^{a^3}+...+x^{a^n}$

Note that $k$ is a power to $a$ not $x$.

It is simple to find a closed form for $\sum_{k=0}^{n} x^{k}$

which is $\frac {1-x^n}{1-x}$

but I tried to find a closed form for $\sum_{k=0}^{n} x^{a^k}$ without any useful results.

asked Aug 07 '16 at 22:28

Pentapolis

http://math.stackexchange.com/questions/404510/finding-the-sum-xx2x4x8x16-cdots might be of use – Peter Huxford Aug 07 '16 at 22:43
I dont think that exist a closed form for this iterated exponential function. – Aug 07 '16 at 23:01
Generally, if there is simple form for the integral $\int_a^bf(t)dt$, there is likely no simple form for the summation $\sum_{k=a}^bf(k)$ – Simply Beautiful Art Aug 07 '16 at 23:35
2

@SimpleArt Why would a simple form for an integral imply no simple form for a summation? Is there some sort of justification for this claim? – Brevan Ellefsen Aug 08 '16 at 01:12
@BrevanEllefsen Er, I meant the opposite, that if there is no simple integral, most likely no simple summation. – Simply Beautiful Art Aug 08 '16 at 11:40

0 Answers0