Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = cos x, a = π/2

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Identify the function and the center point: here, the function is \(f(x) = \cos x\) and the center is \(a = \frac{\pi}{2}\).

Recall the Taylor series formula 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 \(x = a\).

Calculate the first few derivatives of \(f(x) = \cos x\) and evaluate them at \(x = \frac{\pi}{2}\): - \(f(x) = \cos x\) so \(f\left(\frac{\pi}{2}\right) = 0\) - \(f'(x) = -\sin x\) so \(f'\left(\frac{\pi}{2}\right) = -1\) - \(f''(x) = -\cos x\) so \(f''\left(\frac{\pi}{2}\right) = 0\) - \(f'''(x) = \sin x\) so \(f'''\left(\frac{\pi}{2}\right) = 1\)

Write out the first three nonzero terms of the Taylor series using the derivatives found: \[f(x) \approx f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots\] Substitute the values and keep only the first three nonzero terms.

Express the Taylor series in summation notation: \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}\left(\frac{\pi}{2}\right)}{n!} (x - \frac{\pi}{2})^n,\] where you can identify the pattern of nonzero terms from the derivatives calculated.

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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. Each term involves the nth derivative evaluated at the center point, multiplied by (x - a)^n and divided by n!. This allows approximation of functions near the point a.

Derivatives of Trigonometric Functions

Understanding the derivatives of cosine is essential, as they follow a repeating cycle: cos x, -sin x, -cos x, sin x, then back to cos x. Evaluating these derivatives at the center point helps determine the coefficients of the Taylor series terms.

Summation Notation for Series

Summation notation concisely expresses infinite or finite sums using the sigma symbol (∑). Writing the Taylor series in summation form involves identifying the general term formula, including factorial denominators and powers of (x - a), to represent the series compactly.

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.ƒ(x) = cos x, a = π/2