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How to compute $$\sum_{k=1}^{180}\cos\left(\frac{2\pi}{k}\right)$$

I have tried several ways to do it (including euler formula, etc), but all failed. The answer given by wolframalpha appeared to be a bad irrational number...

Edited: The source of the problem is a test from our school. Maybe there is a typo on the upper bound $180$. How if the problem changes to $\sum_{k=1}^{180^\circ}\cos\left(\frac{2\pi}{k}\right)$? Is there a chance to have a closed form?

Shara
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    What makes you think there is a closed form? – saulspatz Apr 11 '21 at 11:05
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    What's the source of this problem? The summand and range for index both look strange. – Macavity Apr 11 '21 at 11:49
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    Pity it's not something like $\cos(2\pi k/180)$. Anyway, here's your sum to 100 places, computed using mpmath: 171.4640044505613428214098838495749812772666106357843039716361604645700754843936911252650831636893844 – PM 2Ring Apr 11 '21 at 11:58
  • Strange. The $2\pi$ suggests we're working in radians, but the $180$ suggests we're working in degrees. – Gerry Myerson Apr 11 '21 at 12:51
  • I mean, you can loosely approximate if you like. Just write down all the cos values you know first, and then approximate other terms with $k\ge 5$ as $1$. I get $174.07$ which is not bad. – Vishu Apr 11 '21 at 12:59
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    @GerryMyerson Editted! – Shara Apr 11 '21 at 13:02
  • We will all be interested in seeing the solution your teacher gives to this test question. I hope you will post it here, or ask your teacher to post it for you. – Gerry Myerson Apr 11 '21 at 23:00

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$$\sum_{k=1}^{180}\cos\left(\frac{2\pi}{k}\right)=\sum_{k=1}^{6}\cos\left(\frac{2\pi}{k}\right)+\sum_{k=7}^{180}\cos\left(\frac{2\pi}{k}\right)$$ $$\sum_{k=1}^{6}\cos\left(\frac{2\pi}{k}\right)=\frac{1}{4} \left(\sqrt{5}-1\right)$$

Now, using Taylor series $$\cos\left(\frac{2\pi}{k}\right)=1-\frac{2 \pi ^2}{k^2}+\frac{2 \pi ^4}{3 k^4}-\frac{4 \pi ^6}{45 k^6}+\frac{2 \pi ^8}{315 k^8}-\frac{4 \pi ^{10}}{14175 k^{10}}+O\left(\frac{1}{k^{12}}\right)$$ You then have a long analytical formula and, numerically

$$\sum_{k=1}^{180}\cos\left(\frac{2\pi}{k}\right)=171.4640044498\cdots$$ while the exact value is $171.4640044506\cdots$

Claude Leibovici
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