Write the Taylor series for f(x)=cos(x)f(x) = \cos(x)f(x)=cos(x) centered at x=π2x = \frac{\pi}{2}x=2π.
∑n=0∞(−1)n(2n)!(x−π2)2n\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} \left(x - \frac{\pi}{2}\right)^{2n}∑n=0∞(2n)!(−1)n(x−2π)2n
∑n=0∞(−1)n+1(2n+1)!(x−π2)2n+1\sum_{n=0}^{\infty} \frac{(-1)^{n+1}}{(2n+1)!} \left(x - \frac{\pi}{2}\right)^{2n+1}∑n=0∞(2n+1)!(−1)n+1(x−2π)2n+1
−∑n=0∞(−1)n(2n+1)!(x−π2)2n+1-\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \left(x - \frac{\pi}{2}\right)^{2n+1}−∑n=0∞(2n+1)!(−1)n(x−2π)2n+1
∑n=0∞(−1)n(2n+1)!(x−π2)2n+1\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \left(x - \frac{\pi}{2}\right)^{2n+1}∑n=0∞(2n+1)!(−1)n(x−2π)2n+1