Problem Statement

In Exercises 7 – 12, find a formula for the nth term of the Taylor series of \(f(x)\), centered at \(c\), by finding the coefficients of the first few powers of \(x\) and looking for a pattern. (The formulas for several of these are found in Key Idea 8.8.1; show work verifying these formulas.)

Exercise 7

  • Function: \(f(x) = \cos(x)\)
  • Center: \(c = \frac{\pi}{2}\)

To find the Taylor series for \(f(x) = \cos(x)\) centered at \(c = \frac{\pi}{2}\), we start by computing the derivatives of \(f(x)\) and evaluate them at \(c\). The Taylor series expansion is given by:

\[ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(c)}{n!}(x - c)^n \]

Function and Derivatives

  1. Zeroth derivative (Function itself): \[ f(x) = \cos(x) \quad \Rightarrow \quad f\left(\frac{\pi}{2}\right) = 0 \]

  2. First derivative: \[ f'(x) = -\sin(x) \quad \Rightarrow \quad f'\left(\frac{\pi}{2}\right) = -1 \]

  3. Second derivative: \[ f''(x) = -\cos(x) \quad \Rightarrow \quad f''\left(\frac{\pi}{2}\right) = 0 \]

  4. Third derivative: \[ f'''(x) = \sin(x) \quad \Rightarrow \quad f'''\left(\frac{\pi}{2}\right) = 1 \]

Identifying the Pattern

Observing the derivatives at \(c = \frac{\pi}{2}\), we see a pattern emerging. The derivatives of \(\cos(x)\) evaluated at \(\frac{\pi}{2}\) alternate between 0 and \(\pm 1\). The non-zero terms only appear at odd derivatives. Specifically:

  • Even derivatives (\(f^{(2k)}\)) are 0.
  • Odd derivatives (\(f^{(2k+1)}\)) follow the pattern \((-1)^k\).

Taylor Series Formula

Given the observed pattern, we can express the Taylor series of \(\cos(x)\) centered at \(\frac{\pi}{2}\) as:

\[ \cos(x) = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} (x - \frac{\pi}{2})^{2k+1} \]

This series converges to \(\cos(x)\) for all \(x\) and succinctly captures the oscillatory behavior of the cosine function around \(\frac{\pi}{2}\).