I know that $\displaystyle\sin x = \sum_{k=0}^{\infty}\frac{\left( -1 \right) ^kx^{2k+1}}{(2k+1)!}$
But how to get transformation to get Series about x?
2026-03-29 22:08:18.1774822098
How to get Taylor Series of $\sin \frac{x}{1-x}$
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comment
From Maple, I get $$\sin\frac{x}{1-x} = x+{x}^{2}+{\frac {5}{6}}{x}^{3}+{\frac {1}{2}}{x}^{4}+{\frac {1}{120} }{x}^{5}-{\frac {5}{8}}{x}^{6}-{\frac {6931}{5040}}{x}^{7}-{\frac { 1591}{720}}{x}^{8}-{\frac {224179}{72576}}{x}^{9}-{\frac {31987}{8064} }{x}^{10}-{\frac {27323293}{5702400}}{x}^{11}-{\frac {19986991}{ 3628800}}{x}^{12}-{\frac {1508675351}{249080832}}{x}^{13}-{\frac { 3055361231}{479001600}}{x}^{14}-{\frac {8385570334051}{1307674368000}} {x}^{15}-{\frac {16115825351}{2641766400}}{x}^{16}-{\frac { 1916072623603199}{355687428096000}}{x}^{17}-{\frac {88344037902079}{ 20922789888000}}{x}^{18}-{\frac {12475149135090259}{4865804016353280}} {x}^{19}+O \left( {x}^{20} \right) $$ So, can we "recognize" these terms to guess a formula? Can we even say that all subsequent coefficients are negative?