What is the Maclaurin series for arcsin(x)\arcsin(x)arcsin(x)?\\Hint: ddx[arcsin(x)]=(1−x2)−1/2\frac{d}{dx}[\arcsin(x)] = (1-x^2)^{-1/2}dxd[arcsin(x)]=(1−x2)−1/2 and use the binomial series.
x+x33+x55+x77+⋯x + \frac{x^3}{3} + \frac{x^5}{5} + \frac{x^7}{7} + \cdotsx+3x3+5x5+7x7+⋯
x−x36+3x540−⋯x - \frac{x^3}{6} + \frac{3x^5}{40} - \cdotsx−6x3+403x5−⋯
1+x2+3x28+5x316+⋯1 + \frac{x}{2} + \frac{3x^2}{8} + \frac{5x^3}{16} + \cdots1+2x+83x2+165x3+⋯
x+x36+3x540+5x7112+⋯x + \frac{x^3}{6} + \frac{3x^5}{40} + \frac{5x^7}{112} + \cdotsx+6x3+403x5+1125x7+⋯