The geometric series can be solve efficiently with
$\sum_{k=0}^{n} r^k = \frac{1 - r^{n+1}}{1 - r}$
Is the a series or approximation for the more complex case:
$\sum_{k=0}^{n} r^{k \sqrt{1 + bk^2}} = ?$
or for
$\sum_{k=0}^{n} r^{k^m} = ?$
r, n, b, m are constants
Edit 1: If no exact solutions exists, I will be happy with an approximation that is quicker than direct summation.
$|r| < 1$ r is complex
m is real number between 1 and 2
b is a real number between 0 and 1
n is less than 1024
The exponent $f(k)$ does need to be $k \sqrt{1 + bk^2}$ or $k^m$ but ideally f should be monotonically increasing and $n < f(n) < \sqrt{2}n^2$ and f'(k) > 0, f"(k) > 0. Solution for such similar functions will be helpful, like maybe there is an exact solution exist for polynomials, I don't know.
I know one approximation is to linearize:
$\sum_{k=0}^{n_0}r_0^{a_0 k} + \sum_{k=(n_0+1)}^{n_1}r_1^{a_1 k} + ... + \sum_{k=(n_{i-1})}^{n}r_i^{a_i k}$
and use the closed form of each term.
Linearizing creates a function f(k) that has the properties I am looking for except that f"(k) is not > 0 everywhere from k = 0 to 1024. Is there a another approximation?