I want to find a smooth function $f$ such that for all positive integers $x$, $f(x)=g(x)$ where $g(x)$ is given below. In less mathy terms I want a smooth version of $g(x)$ constrained by its integer inputs. $$g(x) = \sum_{i=0}^x a^x$$ I've been playing around with this on Desmos by plotting $g(x)$ and the formula for a geometric sum $\frac{1-a^{x+1}}{1-a}$ which seems like it could be close to an answer but other than that I have no idea where to go from here. I've thought about trying to differentiate but again, don't know how to do that with this function. Any help or suggestions are welcome.
2026-03-27 22:30:55.1774650655
Find smooth version of $g(x) = \sum_{i=0}^x a^x$
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