Use the Taylor series given in Key Idea 8.8.1 to create the Taylor series of the given functions:
\[ \cos(x^2) \]
From Key Idea 8.8.1: The standard Taylor series expansion for \(\cos x\) is: \[ \cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} \] By substituting \(x^2\) for \(x\), we derive the series for \(\cos(x^2)\): \[ \cos(x^2) = \sum_{n=0}^{\infty} (-1)^n \frac{(x^2)^{2n}}{(2n)!} \] Simplifying the powers of \(x\), we obtain: \[ \cos(x^2) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{4n}}{(2n)!} \]
With interval of convergence \((-\infty,\infty)\)