We all know what the sum of a geometric series is $1 + x + x^2 + x^3 + ... + x^k = \frac{x^{k+1} - 1}{x - 1}$ .
I was wondering if similar formulas exist in case the exponents form some other sequence, for example are squares or cubes of positive integers?
On
This question is not very meaningful unless you define what you mean by "closed form".
Suppose "closed form" means a rational function in $x$ and $x^n$, as in your example for $\sum_i x^i$: $$R(x,x^n) = \frac{p(x,x^n)}{q(x,x^n)} = \sum_{i=0}^{n}x^{i^2}.$$ Then you know that when $|x|<1$ you can take the limit as $n\to\infty$ and get $$ R(x,0) = \lim_{n\to\infty} R(x,x^n) = \sum_{i\geq0} x^{i^2} = \frac12 + \frac12\vartheta_3(0,x), $$ where $\vartheta_3$ is a Jacobi theta function. But the right-hand side is not a rational function, which the left-hand side is, so such a rational function $R(x,x^n)$ cannot exist.
I'm not sure why someone downvoted this question. Perhaps said person was annoyed by the tag 'number theory', which ostensibly it isn't, but there is an interesting irony here.
I'm pretty sure there's no closed form for $\sum_{i=0}^n x^{i^2}$, but that doesn't mean this type of summation isn't important in some ways. Letting $n$ tend to $\infty$, such series form what is called a Jacobi theta function. Theta functions occupy a prominent place in modern-day number theory; the Wikipedia article gives some indication of this, and it would probably be off-topic to go much into it here.
The theta functions are infinite series, but finite series of your form come up in the study of Gauss sums, particularly quadratic Gauss sums, which are also important in number theory (e.g., for the study of quadratic reciprocity). Again, I'll just point to the linked article for further information.