Question

Show that the MacLaurin's series for f (z) =sin z is given by

z z 3 s "' (- I)" z211+J

f(z) =z--+- ....+L ( ) +...

Answer 1

$\displaystyle \text{The maclaurin's series is given by}\\ f(z) = f(0) + zf'(0) + \frac{z^2}{2}f''(0) + \frac{z^3}{6}f'''(0)+\frac{z^4}{24}f^{iv}(0)+\frac{z^5}{120}f^{v}(0)+\cdots\\ \text{Where}\\ f(0) = sin0=0\\ f'(z) = cosz, \qquad f'(0) = 1\\ f''(z) = -sinz, \qquad f''(0) =0\\ f'''(z) = -cosz, \qquad f'''(0) =-1\\ f^{iv}(z) = sinz, \qquad f^{iv}(0) =0\\ f^{v}(z) = cosz, \qquad f^{v}(0) =1\\ \text{Hence, the maclaurin's series is given by}\\ f(z) = z-\frac{z^3}{6}+\frac{z^5}{120}+ \cdots$