Taylor series


b. Write the power series using summation notation.


f(x)=sin x, a = π/2

Verified step by step guidance

1. Recall the Taylor series formula for a function \( f(x) \) centered at \( a \): \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \] where \( f^{(n)}(a) \) is the \( n \)-th derivative of \( f \) evaluated at \( a \).
2. Identify the function and center: here, \( f(x) = \sin x \) and \( a = \frac{\pi}{2} \). We need to find the derivatives of \( \sin x \) evaluated at \( x = \frac{\pi}{2} \).
3. Calculate the first few derivatives of \( \sin x \) and evaluate them at \( x = \frac{\pi}{2} \): - \( f(x) = \sin x \) - \( f'(x) = \cos x \) - \( f''(x) = -\sin x \) - \( f^{(3)}(x) = -\cos x \) - \( f^{(4)}(x) = \sin x \) Evaluate each at \( x = \frac{\pi}{2} \): - \( f\left(\frac{\pi}{2}\right) = 1 \) - \( f'\left(\frac{\pi}{2}\right) = 0 \) - \( f''\left(\frac{\pi}{2}\right) = -1 \) - \( f^{(3)}\left(\frac{\pi}{2}\right) = 0 \) - \( f^{(4)}\left(\frac{\pi}{2}\right) = 1 \)
4. Notice the pattern in the derivatives evaluated at \( a = \frac{\pi}{2} \): the values cycle through \( 1, 0, -1, 0, 1, \ldots \). This pattern will help simplify the summation.
5. Write the Taylor series in summation notation using the pattern found: \[ \sin x = \sum_{n=0}^{\infty} \frac{f^{(n)}\left(\frac{\pi}{2}\right)}{n!} (x - \frac{\pi}{2})^n \] Substitute the values of the derivatives into the summation to express the power series explicitly.

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. For a function f(x) centered at a point a, it is expressed as f(x) = Σ (f⁽ⁿ⁾(a)/n!) (x - a)ⁿ, where n! is factorial and f⁽ⁿ⁾(a) is the nth derivative at a.

Derivatives of sin(x)

The derivatives of sin(x) follow a repeating cycle every four derivatives: sin(x), cos(x), -sin(x), -cos(x), then back to sin(x). Evaluating these derivatives at x = π/2 simplifies the coefficients in the Taylor series, as sin(π/2) = 1 and cos(π/2) = 0.

Summation Notation for Power Series

Summation notation (Σ) compactly expresses infinite series by indicating the general term and the index of summation. Writing the Taylor series in summation form involves identifying the pattern of coefficients and powers of (x - a), allowing a concise representation of the entire series.