Infinite equation

Calculus Level 5

\[ \mathscr{E} = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} +\cdots \]

For how many of the following value(s) of \( \mathscr{E} \) will there be a real number value of \( x \) that satisfies the above equation?

\[ -2, \ -\dfrac{3}{2}, \ -1, \ -\dfrac{1}{2}, \ 0, \ \dfrac{1}{2}, \ 1, \ \dfrac{3}{2}, \ 2 \]

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