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I have a table that where the elements are the results of Solve:

table = 
  Table[
    Solve[
      Sum[(2 j π Sin[(2 j π  x)/(r + 1)])/(r + 1)^2, {j, 0, r}] == 0 && 
      0 < x <= 1],
    {r, 25}]

My issue is that, from r = 4 onwards, the results are not expressed in fractional form, but as approximations in decimal form. I assume it's because the fractions would become long or complicated, but I want the expanded fractional form regardless of complexity.

How do I force Mathematica to express the results as fractions?

EDIT

My main aim is to derive a systematic formula for the zeroes of the equation within that range of x. Unfortunately, I'm a total amateur, and when I see things like 6 ArcTan[Sqrt[Root[55 - 396 #1 + 594 #1^2 - 220 #1^3 + 15 #1^4 &, 1]]] I kind of freeze. What does something like #1 even mean? I thought I saw a pattern developing in the first few outputs. But then I lost it because I can't interpret what Mathematica is trying to tell me.

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Richard Burke-Ward
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    When I run your code I get answers in terms of Root objects, which are precise objects but involve roots of polynomials. – SPPearce Sep 07 '18 at 09:19
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    To add to the comment of @KraZug: What version of Mathematica are you on? I get Root objects as well, not decimal approximations. – Sjoerd Smit Sep 07 '18 at 09:21
  • OK, I probably wasn't clear enough. Wasn't trying to be cryptic, just brief and to the point. Would be v grateful if you could look at my edit of the OP, and help me further. – Richard Burke-Ward Sep 07 '18 at 09:27
  • And thanks for the formatting help :-) – Richard Burke-Ward Sep 07 '18 at 09:31
  • Richard, have a look into the documentation of Function in order to learn about more about these somewhat cryptic things like #, #1,##, and &. – Henrik Schumacher Sep 07 '18 at 09:33
  • MMA11 home use, BTW. – Richard Burke-Ward Sep 07 '18 at 09:33
  • Root objects are a way of expressing roots of polynomial equations. You can use ToRadicals to get the cubic and quartic ones to explicit expressions, but only up to n=4. Incidentally, Mathematica seems to give a closed form for the sum with arbitrary r. – SPPearce Sep 07 '18 at 09:45
  • Hi Henrik. (Again! You are such a helpful person! Vielen dank.) So, in my case, #1 means 'parameter 1'... Which is...x...? Sorry to be a bit slow. I have an English degree, but the last time I did any serious mathematics, I was 18 - a long time ago... – Richard Burke-Ward Sep 07 '18 at 09:52
  • Hi KraZug, how did you get to the closed form? It looks to me like it's convergent, but I don't know how to get to the actual final expression... – Richard Burke-Ward Sep 07 '18 at 09:53
  • Also: KraZug - I very much appreciate your input too. I only singled out Heinrik because he's answered a few other questions I posed :-) – Richard Burke-Ward Sep 07 '18 at 09:55
  • D'oh! #1 must be r... I think...? – Richard Burke-Ward Sep 07 '18 at 10:06
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    Related: https://mathematica.stackexchange.com/questions/13767/how-do-i-work-with-root-objects – Michael E2 Sep 07 '18 at 11:09
  • You might want to have a look into a nice online book of L. Shifrin. You may find it here: http://www.mathprogramming-intro.org/book/node50.html The chapter 4.11 is what you need. – Alexei Boulbitch Sep 07 '18 at 12:43

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