Question: Find the Maclaurin series for \( f(x) \) using the definition of a Maclaurin series. [Assume that…

Find the Maclaurin series for \( f(x) \) using the definition of a Maclaurin series. [Assume that \( f \) has a power series expansion. Do not show that \( R = \infty \).]

\[ f(x) = \cos(x) \]

\[ f(x) = \sum_{n=0}^{\infty} \left( \boxed{\phantom{xxxx}} \right) \]

Find the associated radius of convergence \( R \).

\[ R = \boxed{\phantom{xxxx}} \]

Solution

To find the Maclaurin series for \( f(x) = \cos(x) \): \[ f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \] Next, determine the radius of convergence: Using the ratio test: \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(-1)^{n+1} x^{2(n+1)}}{(2(n+1))!} \cdot \frac{(2n)!}{(-1)^n x^{2n}} \right| \] \[ = \lim_{n \to \infty} \frac{x^2}{(2n+1)(2n+2)} = 0 \] Since the limit is 0 for all real \( x \), the radius of convergence is infinite. \[ R = \infty \]