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How would I simplify these expressions into power series notation in mathematica?

For example, in a 3 body system of the earth moon and sun. Where masses are

Sun=M
Earth=m1
Moon=m2

Distance from sun to earth = r1
Distance from earth to moon= r2
Distance from sun to moon: r3=r2-r1

enter image description here

For example

\!\(
\*SubsuperscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]
\*FractionBox[\(1\), 
SuperscriptBox[\(i\), \(6\)]]\)

which gives

\[Pi]^6/945
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closed as unclear what you're asking by Michael Seifert, Szabolcs, m_goldberg, MarcoB, Alex Trounev Oct 23 at 19:21

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ What do you mean by "sum notation"? Can you edit your question to give an example of what you're looking for using LaTeX? $\endgroup$ – Michael Seifert Oct 21 at 17:58
  • $\begingroup$ Apologies for the delay, i've updated the question with an example. What im struggling to see is to how i can show these equations in the above mentioned sum notation. Thanks for your reply $\endgroup$ – Luke4737 Oct 22 at 6:01
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If you want to work with these equations

eqs = Table[
 r[k]''[t] == -G Sum[
  m[j] (r[k] - r[j])/Norm[r[k] - r[j]]^3,
  {j, Complement[{1, 2, 3}, {k}]}
 ], 
 {k, 3}
];
eqs /. {r[k_]''[t] :> Subscript[Overscript[r, ".."], k], r[k_] :> Subscript[r, k], m[k_] :> Subscript[m, k]} // Column // TeXForm

$\begin{array}{l} \overset{\text{..}}{r}_1=-G \left(\frac{m_2 \left(r_1-r_2\right)}{\left\| r_1-r_2\right\| {}^3}+\frac{m_3 \left(r_1-r_3\right)}{\left\| r_1-r_3\right\| {}^3}\right) \\ \overset{\text{..}}{r}_2=-G \left(\frac{m_1 \left(r_2-r_1\right)}{\left\| r_2-r_1\right\| {}^3}+\frac{m_3 \left(r_2-r_3\right)}{\left\| r_2-r_3\right\| {}^3}\right) \\ \overset{\text{..}}{r}_3=-G \left(\frac{m_1 \left(r_3-r_1\right)}{\left\| r_3-r_1\right\| {}^3}+\frac{m_2 \left(r_3-r_2\right)}{\left\| r_3-r_2\right\| {}^3}\right) \\ \end{array}$

If you're looking for a programmatic way to typeset these equations in summation notation

typeset = {r[k_]''[t] :> Subscript[Overscript[r, ".."], k], r[k_] :> Subscript[r, k], m[k_] :> Subscript[m, k]};

general = HoldForm[r[k]''[t] == -G Sum[m[j] (r[k] - r[j])/Norm[r[k] - r[j]]^3, j != k]] /. typeset // TraditionalForm

$\overset{\text{..}}{r}_k=-G \sum_{j\neq k} \frac{m_j \left(r_k-r_j\right)}{\left\| r_k-r_j\right\| {}^3}$

general /. List /@ Thread[k -> Range[3]] // Column // TraditionalForm

$\begin{array}{l} \overset{\text{..}}{r}_1=-G \sum _{j\neq 1} \frac{m_j \left(r_1-r_j\right)}{\left\| r_1-r_j\right\| {}^3} \\ \overset{\text{..}}{r}_2=-G \sum _{j\neq 2} \frac{m_j \left(r_2-r_j\right)}{\left\| r_2-r_j\right\| {}^3} \\ \overset{\text{..}}{r}_3=-G \sum _{j\neq 3} \frac{m_j \left(r_3-r_j\right)}{\left\| r_3-r_j\right\| {}^3} \\ \end{array}$

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