Using the Maclaurin series cos(x)=∑n=0∞(−1)nx2n(2n)!\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}cos(x)=∑n=0∞(2n)!(−1)nx2n, find the series for cos(x2)\cos(x^2)cos(x2).
∑n=0∞(−1)nx2n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}∑n=0∞(2n)!(−1)nx2n
∑n=0∞(−1)nx4n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{4n}}{(2n)!}∑n=0∞(2n)!(−1)nx4n
1−x42!+x84!−x126!+⋯1 - \frac{x^4}{2!} + \frac{x^8}{4!} - \frac{x^{12}}{6!} + \cdots1−2!x4+4!x8−6!x12+⋯
∑n=0∞(−1)nx4n(4n)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{4n}}{(4n)!}∑n=0∞(4n)!(−1)nx4n