I am doing some nested regular polygon fractals for fun. I need to solve this sum for the $n$th partial sum to move forward with the exploration:
$$\sum_{k=3}^n\frac{1}{\cos\left(\frac{\pi}{k}\right)-\cos^2\left(\frac{\pi}{k}\right)}$$
The partial sum will tell me the sum of all the sides for each polygon with $n$ sides. I arrived at this problem after substituting the closed-form of an infinite geometric series with $a$ equal to $\dfrac{1}{\cos\left(\frac{\pi}{n}\right)}$ and $r$ equal to $\cos\left(\dfrac{\pi}{n}\right)$. I am completely lost at this stage.
I do not know if a closed form would exist.
What you could do is to write $$S_n=\sum_{k=3}^n\frac{1}{\cos\left(\frac{\pi}{k}\right)-\cos^2\left(\frac{\pi}{k}\right)}=\left(12+2 \sqrt{2}+\frac{8}{\sqrt{3}}+2 \sqrt{5}\right)+\frac{1}{2}\sum_{k=7}^n \csc ^2\left(\frac{\pi }{2 k}\right) \sec \left(\frac{\pi }{k}\right)$$ now develop the summand as a Taylor series. This would give $$\csc ^2\left(\frac{\pi }{2 k}\right) \sec \left(\frac{\pi }{k}\right)=\frac{4 k^2}{\pi ^2}+\frac{7}{3}+\frac{61 \pi ^2}{60 k^2}+\frac{631 \pi ^4}{1512 k^4}+\frac{7321 \pi ^6}{43200 k^6}+\frac{91411 \pi ^8}{1330560 k^8}+O\left(\frac{1}{k^{10}}\right)$$ Now, you can sum from $7$ to $n$ to get for $$T_n=\sum_{k=7}^n \csc ^2\left(\frac{\pi }{2 k}\right) \sec \left(\frac{\pi }{k}\right)$$ a quite simple expression where appear a constant, a cubic polynomial in $n$ and a few generalized harmonic numbers. For sure, if $n$ is large, you can expand again the result using the asymptotics of the generalized harmonic numbers.
This would give, as an approximation for large $n$ $$S_n=-\frac{361}{498}+\frac{2 n^3}{3 \pi ^2}+\frac{n^2}{\pi ^2}+\left(\frac{7}{6}+\frac{1}{3 \pi ^2}\right) n-\frac{61 \pi ^2}{120 n}+\frac{61 \pi ^2}{240 n^2}-\frac{\pi ^2 \left(3843+3155 \pi ^2\right)}{45360 n^3}+\frac{631 \pi ^4}{6048 n^4}+O\left(\frac{1}{n^5}\right)$$
For a small value such as $n=10$, the exact value is $88.4757$ while the approximation gives $88.4759$. For $n=100$, the exact value is $68679.9368097$ while the approximation gives $68679.9368107$.