Find the Taylor series for f(x)=sin(x2)f(x) = \sin(x^2)f(x)=sin(x2) centered at x=0x=0x=0.
∑n=0∞(−1)nx2n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}∑n=0∞(2n)!(−1)nx2n
∑n=0∞(−1)nx4n+2(2n+1)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{4n+2}}{(2n+1)!}∑n=0∞(2n+1)!(−1)nx4n+2
∑n=0∞(−1)nx4n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{4n}}{(2n)!}∑n=0∞(2n)!(−1)nx4n
∑n=0∞(−1)nx2n+2(2n+1)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+2}}{(2n+1)!}∑n=0∞(2n+1)!(−1)nx2n+2