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I like to do the following; enter image description here

but the representation as sum it does not match with the taylor series how could it be done?? Thanks in advance

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    $\begingroup$ You mistakenly used only one index in the powers of x and y in the sum. $\endgroup$
    – Bob Hanlon
    Feb 26 at 21:31
  • $\begingroup$ @BobHanlon (+1) for the comment and sorry. I just saw your comment as I edited the answer. I am happy to delete mine if you would like to write one. $\endgroup$
    – kcr
    Feb 26 at 21:34
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    $\begingroup$ @DiSp0sablE_H3r0 - Don't worry about it. $\endgroup$
    – Bob Hanlon
    Feb 26 at 21:36

2 Answers 2

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You need to use two distinct indices in the sums.

srs = Normal[Series[ArcSin[x + y]^2, {x, 0, 3}, {y, 0, 3}]] // Expand;
g = CoefficientList[srs, {x, y}];

Sum[g[[k + 1, j + 1]] x^k y^j, {k, 0, 3}, {j, 0, 3}]

which gives back

x^2 + 2 x y + (4 x^3 y)/3 + y^2 + 2 x^2 y^2 + (4 x y^3)/3 + (
 32 x^3 y^3)/9
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Maybe

ser = Series[ArcSin[x + y]^2, {x, 0, 3}, {y, 0, 3}] // Normal // Expand

g = CoefficientList[ser, {x, y}, 4]

(* {{0, 0, 1, 0}, {0, 2, 0, 4/3}, {1, 0, 2, 0}, {0, 4/3, 0, 32/9}} *)
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