f(x) = sin(4x)
f' = 4 cos(4x)
f'' = -16 sin(4x)
f''' = -64 cos(4x)
f⁽⁴⁾ = 256 sin(4x)
f⁽⁵⁾ = 1024 cos(4x)
The 4-th order Taylor series expansion is
f(x+h) = f(x) + hf'(x) + (h²/2!)f''(x) + (h³/3!)f'''(x) + (h⁴/4!)f⁽⁴⁾(x) + ...
The Maclaurin series is obtained by setting x = 0.
Note that sin(0) = 0 and cos(0) = 1.
The non zero terms are
f(h) = 4h - (4h)³/3! + (4h)⁵/5! - (4h)⁷/7! + ...
Answer:
[tex]f(x) = 4x - \frac{(4x)^{3}}{3!} + \frac{(4x)^{5}}{5!} - \frac{(4x)^{7}}{7!} + \, ...,[/tex]
