Using the definition, find the Taylor series of sin(x) centered at x=pi/2

Question

Using the definition, find the Taylor series of sin(x) centered at x=pi/2. Show the first five derivatives and how they are used to find the coefficients of the series. Write your final answer in sigma notation. You do not need to specify the interval of convergence. (If correct, the series converges for all real x.)

Answer 1

f(x) = sinx; f(π/2) = sinπ/2 = 1; f'(x) = cosx, f'(π/2)= cosπ/2 = 0, f''(x) = - sinx; f''(π/2) = - sinπ/2 = - 1;

f'''(x) = - cosx; f'''(π/2) = - cosπ/2 = 0; f⁽⁴⁾(x) = sinx; f⁽⁴⁾(π/2) = sin(π/2) = 1; f⁽⁵⁾(x) = cosx, f⁽⁵⁾(π/2) = cosπ/2 = 0.

So, sinx = 1 -1/2!(x - π/2)² + 1/4!(x - π/2)⁴ - ... + (-1)ⁿ1/(2n)!(x - π/2)²ⁿ + ... = ∑₀∞(-1)ⁿ(x - π/2)²ⁿ/(2n)!