91Ó°ÊÓ

The cosine function, denoted as \( \cos x \), is a fundamental trigonometric function characterized by its periodicity. This periodic behavior means that it repeats its values in regular intervals of \( 2\pi \). In the context of Taylor series, especially when centered around zero, the cosine function exhibits a series expansion that comprises only even powers of \( x \). Based on this behavior, the Taylor series for \( \cos x \) can be represented as: \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \] This series is derived from the derivatives of \( \cos x \), each alternating in sign and factorial growth in the denominator. Its symmetry and alternating series nature play a crucial role in approximating cosine in various intervals and are integral in the estimation of functions using Taylor series.

In the context of Taylor series, finding the maximum value of the absolute value of the \((n+1)\)-th derivative, \(f^{(n+1)}(z)\), on a specified interval is crucial for estimating the remainder term. For the cosine function on the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\), both \( |\cos x| \) and \( |\sin x| \) reach their maximum value of 1. This is because the sine and cosine functions oscillate between -1 and 1. Consequently, when estimating the remainder, the maximum value \( M \) is taken as 1. This approach ensures that the remainder estimate for the associated Taylor series expansion is as tight as possible.

• The periodic and bounded nature of sine and cosine eases the process of finding maximum values.
• Being consistent in estimating \( M \) across intervals helps achieve better approximations in Taylor series.

Calculating derivatives of trigonometric functions like cosine follows a predictable cycle. Starting from the function itself, \( f(x) = \cos x \), deriving step-by-step results in the following sequence:

• \( f'(x) = -\sin x \)
• \( f''(x) = -\cos x \)
• \( f'''(x) = \sin x \)
• \( f^{(4)}(x) = \cos x \)

This cycle follows every four derivatives, suggesting that the pattern of derivatives repeats after every fourth step. Recognizing this cycle simplifies calculations, especially when dealing with higher-order derivatives. These derivatives are utilized in the Taylor series to approximate functions close to a given point, in this case, around \( x = 0 \). Note that the sign alternates and the derivatives switch between sine and cosine every two steps.

Solving inequalities is pivotal in concluding Taylor series applications, particularly to confirm the acceptable range for the series remainder. For this exercise, the goal is to ensure the inequality \[ \left(\frac{\pi}{2}\right)^{n+1} \leq \frac{1000}{(n+1)!} \] holds true. This process involves substituting trial values for \( n \) to find the smallest integer \( n \) that makes the inequality valid.

• Start with initial simple estimates for \( n \).
• Iteratively try successive values, increasing \( n \) till the inequality holds.
• This approach guarantees accuracy in defining bounds for Taylor series approximation.

For the specific problem, computations show for \( n = 9 \), drawing values from the inequality results in a valid condition, thus confirming that larger \( n \) values precisely bound the remainder term for \[ |R_n| \leq \frac{1}{1000} \].