Partial sum of $\sec nx$

Question

Does there exist a closed form of $$\sum_{n=1}^n \sec nx$$ that is expressed in elementary terms? $${}{}{}{}{}{}{}{}{}{}{}$$

Answer 1

Admitting that you look for $$S_n=\sum_{k=1}^n \sec(k x)$$ if $nx$ is small, you could approximate using $$\sec(x)=1+\frac{x^2}{2}+\frac{5 x^4}{24}+\frac{61 x^6}{720}+\frac{277 x^8}{8064}+O\left(x^{10}\right)$$ and get $$S_n=\sum_{k=1}^n 1+\frac{x^2}{2}\sum_{k=1}^n k^2+\frac{5 x^4}{24}\sum_{k=1}^n k^4+\frac{61 x^6}{720}\sum_{k=1}^n k^6+\frac{277 x^8}{8064}\sum_{k=1}^n k^8+\cdots$$ and use Faulhaber's formulae and get $$S_n=n+\frac{n}{12} \left(2 n^2+3 n+1\right) x^2+\frac{n}{144} \left(6 n^4+15 n^3+10 n^2-1\right) x^4+\frac{61n \left(6 n^6+21 n^5+21 n^4-7 n^2+1\right) }{30240}x^6+\frac{277n \left(10 n^8+45 n^7+60 n^6-42 n^4+20 n^2-3\right) }{725760} x^8+\cdots$$

Using for example $n=20$ and $x=\frac \pi{180}$, the above would give $$20+\frac{287 \pi ^2}{6480}+\frac{361333 \pi ^4}{2519424000}+\frac{1320380441 \pi ^6}{2448880128000000}+\frac{1395698742463 \pi ^8}{634749729177600000000}$$ which is $\approx 20.45163552$ while the exact summation would give $\approx 20.45163645$.

Edit

If, instead of Taylor series, we use a $[4,2]$ Padé approximant built at $x=0$ $$\sec(x)=\frac{1+\frac{7 }{75}x^2+\frac{1}{200}x^4 } {1-\frac{61 }{150}x^2 }$$ which is equivalent to a Taylor series up to $O\left(x^{8}\right)$ we have $$S_n=\sum_{k=1}^n \frac{1+\frac{7 }{75}i ^2x^2+\frac{1}{200}i^4x^4 } {1-\frac{61 }{150}i^2x^2 }$$ and we can get the "surprising" expression $$1815848\, x\, S_n=93750 \sqrt{366} \left(H_{n+\frac{5 \sqrt{\frac{6}{61}}}{x}}-H_{n-\frac{5 \sqrt{\frac{6}{61}}}{x}}+\pi \cot \left(\frac{5 \sqrt{\frac{6}{61}} \pi }{x}\right)\right)-122 (3866 n+9375)\, x-3721\, n (n+1) (2 n+1)\, x^3$$ where appear generalized harmonic numbers.

For the worked example, this would give $\approx 20.45163652$ which is better.

For sure, this could be improved using $[2k,2]$ Padé approximants $(k>2)$ which would lead to similar expressions