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I would like to perform the following taylor expansion in $\zeta$ for a general positive integer n. It works if I tell mathematica n is a given integer, say 3 (see example) but it fails if I leave it as n. How can I get mathematica to perform this computation?

enter image description here

here is the code:

F[x_, \[Zeta]_, n_] := A x^n + B x^(n + 1) - \[Zeta]
G[n_] := Series[
  F[a \[Zeta]^(1/n) - b  \[Zeta]^(2/n) + c  \[Zeta]^(3/n) + 
    d  \[Zeta]^(4/n), \[Zeta], n], { \[Zeta], 0, 2}, 
  Assumptions -> {n \[Element] Integers, n > 0}]
G[n]
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    $\begingroup$ Please do not post un-copyable images. Instead, post code. $\endgroup$
    – David G. Stork
    Commented Jul 21, 2021 at 19:28
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    $\begingroup$ With Table[SeriesCoefficient[F[.....], {\[Zeta], 0, j}], {n, 1, 7}, {j, 0, 2}] // Simplify // MatrixForm i see no systematic. May be you can not find a formula for general n. $\endgroup$
    – Akku14
    Commented Jul 21, 2021 at 20:05

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