Maximizing a function with infinitely many parameters

If I had a function $f$ that was this: $$f\left(a_{0},a_{1},a_{2},a_{3},...a_{\infty},b_{0},b_{1},b_{2},b_{3},...b_{\infty}\right)=\frac{\sum_{k=0}^{\infty}\frac{1}{k+1}\left(\sum_{l=0}^{k}b_{l}\left(k-l+1\right)a_{k-l+1}\right)}{\int_{0}^{1}\sqrt{\left(\sum_{n=0}^{\infty}na_{n}t^{n-1}\right)^{2}+\left(\sum_{n=0}^{\infty}nb_{n}t^{n-1}\right)^{2}}dt}$$ What combination of $a_{0},a_{1},a_{2},a_{3},...a_{\infty},b_{0},b_{1},b_{2},b_{3},...b_{\infty}$ would get me the maximum value of the function? $$$$ Context: I've derived this problem from making a parametric function that has ordinary power series for both x and y coordinates. I then used Green's theorem to make a general formula for the area of the parametric function and put it on top. I used the arc length formula on the function and put it at the bottom. This should mean that if I am able to maximize the function, I will be able to discover the shape that takes up the most area for perimeter (which should be the circle). This came from a proof I saw somewhere where someone proved that the line is the most efficient way to travel between two points using calculus. So, I thought I could try and do the same here, but with a different problem. Overall, the problem had become pretty challenging and more difficult than I had previously imagined, and I am attempting to do some work to simplify both the numerator and the denominator into ordinary power series, which I then can try to maximize the numerator and minimize the denominator, or something similar to that. Unfortunately, I have no idea how to use gradient descent on a function with infinitely many parameters, which is why I've come to you guys to see if I can get any help. I'll give updates if I discover something new that could help. $$$$ Update 1: $f\left(a_{0},a_{1},a_{2},a_{3},...a_{\infty},b_{0},b_{1},b_{2},b_{3},...b_{\infty}\right)=\frac{\sum_{k=0}^{\infty}\frac{1}{k+1}\left(\sum_{l=0}^{k}b_{l}\left(k-l+1\right)a_{k-l+1}\right)}{\sum_{k=0}^{\infty}\frac{p_{k}}{k+1}}$, where $p_{k}=\frac{\sum_{l=0}^{k+2}l\left(k-l+2\right)\left(b_{l}b_{k-l+2}+a_{l}a_{k-l+2}\right)-\sum_{l=1}^{k-1}p_{l}p_{k-l}}{2\sqrt{\left(b_{1}\right)^{2}+\left(a_{1}\right)^{2}}}$ is an equivalent form of the function. However, this new form comes with cons, such as being recursive, so I can't really do much with it yet. I did this by expanding the bottom and solving for a general formula to square root a power series, and then plugging the bottom one in and integrating.