Taylor series and interval of convergence


a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.


f(x) = cosh 3x, a = 0

Verified step by step guidance

1. Recall the definition of the Maclaurin series for a function \(f(x)\) centered at \(a=0\): \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n,\] where \(f^{(n)}(0)\) is the \(n\)-th derivative of \(f\) evaluated at 0.
2. Identify the function: \(f(x) = \cosh(3x)\). We will need to find the derivatives of \(f(x)\) and evaluate them at \(x=0\).
3. Compute the first few derivatives of \(f(x)\): - \(f(x) = \cosh(3x)\) - \(f'(x) = 3 \sinh(3x)\) - \(f''(x) = 9 \cosh(3x)\) - \(f^{(3)}(x) = 27 \sinh(3x)\) - \(f^{(4)}(x) = 81 \cosh(3x)\) and so on.
4. Evaluate these derivatives at \(x=0\): - \(f(0) = \cosh(0) = 1\) - \(f'(0) = 3 \sinh(0) = 0\) - \(f''(0) = 9 \cosh(0) = 9\) - \(f^{(3)}(0) = 27 \sinh(0) = 0\) - \(f^{(4)}(0) = 81 \cosh(0) = 81\)
5. Write the first four nonzero terms of the Maclaurin series using the formula: \[f(x) \approx \sum_{n=0}^{3} \frac{f^{(n)}(0)}{n!} x^n,\] but since odd derivatives are zero, only even terms contribute: \[f(x) \approx f(0) + \frac{f''(0)}{2!} x^2 + \frac{f^{(4)}(0)}{4!} x^4 + \ldots\] Substitute the values found to express these terms explicitly.

Key Concepts

Taylor and Maclaurin Series

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. When centered at zero, it is called a Maclaurin series. Each term involves the nth derivative evaluated at the center, multiplied by (x - a)^n and divided by n!.

Derivatives of Hyperbolic Functions

Understanding the derivatives of hyperbolic functions like cosh(x) is essential. For cosh(3x), derivatives alternate between cosh(3x) and sinh(3x), scaled by powers of 3 due to the chain rule. This pattern helps compute the terms of the Taylor series efficiently.

Interval of Convergence

The interval of convergence is the range of x-values for which the Taylor series converges to the function. For entire functions like cosh(3x), the series converges for all real x, meaning the interval of convergence is (-∞, ∞). This ensures the series accurately represents the function everywhere.