Taylor series and interval of convergence


b. Write the power series using summation notation.


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

Verified step by step guidance

1. Recall the definition of the hyperbolic cosine function: \(\cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}\). This is the Maclaurin series expansion (Taylor series at \(a=0\)) for \(\cosh x\).
2. Since the function is \(f(x) = \cosh 3x\), substitute \$3x\( in place of \)x\( in the series expansion. This gives \)f(x) = \sum_{n=0}^{\infty} \frac{(3x)^{2n}}{(2n)!}$.
3. Simplify the power inside the summation: \((3x)^{2n} = 3^{2n} x^{2n}\). So the series becomes \(f(x) = \sum_{n=0}^{\infty} \frac{3^{2n} x^{2n}}{(2n)!}\).
4. Express the power series explicitly in summation notation centered at \(a=0\) as \(f(x) = \sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!} (x - 0)^{2n}\).
5. This is the power series representation of \(\cosh 3x\) about \(a=0\) in summation notation.

Key Concepts

Taylor Series

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 = 0, it is expressed as the sum of (f^n(0)/n!) * x^n, where f^n(0) is the nth derivative evaluated at zero. This series approximates the function near the center point.

Power Series and Summation Notation

A power series is an infinite series of the form Σ c_n (x - a)^n, where c_n are coefficients and a is the center. Summation notation compactly expresses this infinite sum using the sigma symbol (Σ), indicating the sum over all terms indexed by n. Writing a function as a power series helps analyze and approximate it.

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges to a finite value. It is found by applying convergence tests like the ratio test. Understanding this interval is crucial because the Taylor series only accurately represents the function within this range.