the Key Idea by finding the first few terms of the Taylor series of the given function and identifying a pattern.
\(f(x)=sin(x); c = 0\)
\(f(x) = sin(x)\) \(\quad\) \(f(0) = sin(0) = 0\)
\(f'(x) = cos(x)\) \(\quad\) \(f'(0) = cos(0) = 1\)
\(f''(x) = -sin(x)\) \(\quad\) \(f''(0) = -sin(0) = 0\)
\(f'''(x) = -cos(x)\) \(\quad\) \(f'''(0) = -cos(0) = -1\)
\(f^4(x) = sin(x)\) \(\quad\) \(f^4(0) = sin(0) = 0\)
\(f^5(x) = cos(x)\) \(\quad\) \(f^4(0) = cos(0) = 1\)
\(sin(x)= \frac{f(0)}{0!}(x-0)^0 + \frac{f'(0)}{1!}(x-0)^1 + \frac{f''(0)}{2!}(x-0)^2 + \frac{f'''(0)}{3!}(x-0)^3+ \frac{f^4(0)}{4!}(x-0)^4+ \frac{f^5(0)}{5!}(x-0)^5\) …
\(sin(x)= \frac{0}{0!}(x-0)^0 + \frac{1}{1!}(x-0)^1 + \frac{0}{2!}(x-0)^2 + \frac{-1}{3!}(x-0)^3+ \frac{0}{4!}(x-0)^4+ \frac{1}{5!}(x-0)^5\) …
\(sin(x)= 0 + x+ 0 - \frac{x^3}{3!}+ 0+ \frac{x^5}{5!}\) …
\(sin(x)= x - \frac{x^3}{3!}+ \frac{x^5}{5!}\) …
Verifying the first few terms given in 8.8.1 and the series \(\sum_{n=0}^{\inf}(-1)^n\frac{x^{2n+1}}{(2n+1)!}\)