Taylor series and interval of convergence


c. Determine the interval of convergence of the series.


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

Verified step by step guidance

1. Recall that the Taylor series expansion of \(f(x) = \cosh(3x)\) about \(a = 0\) can be found by using the known series for \(\cosh x\), which is \(\cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}\). Replace \(x\) by \$3x\( to write the series for \)\cosh(3x)\( as \)\sum_{n=0}^{\infty} \frac{(3x)^{2n}}{(2n)!}$.
2. Express the series explicitly as \(\sum_{n=0}^{\infty} \frac{3^{2n} x^{2n}}{(2n)!}\). This is the Taylor series centered at \(a=0\) for \(f(x)\).
3. To determine the interval of convergence, apply the Ratio Test to the general term \(a_n = \frac{3^{2n} x^{2n}}{(2n)!}\). Compute the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
4. Simplify the ratio inside the limit: \(\left| \frac{3^{2(n+1)} x^{2(n+1)}}{(2(n+1))!} \cdot \frac{(2n)!}{3^{2n} x^{2n}} \right| = \left| \frac{3^{2} x^{2}}{(2n+2)(2n+1)} \right|\). Then evaluate the limit as \(n \to \infty\).
5. Since the factorial in the denominator grows faster than any polynomial, the limit \(L\) approaches zero for all real \(x\). Therefore, the series converges for all real numbers, and the interval of convergence is \((-\infty, \infty)\).

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point. For f(x) centered at a, the series is ∑ (f⁽ⁿ⁾(a)/n!) (x - a)ⁿ. It approximates functions like cosh(3x) using polynomial terms around a = 0.

Interval of Convergence

The interval of convergence is the set of x-values for which the Taylor series converges to the function. It is found by applying convergence tests (like the ratio test) to the series terms, determining where the infinite sum is valid and accurately represents the function.

Hyperbolic Cosine Function (cosh x)

The hyperbolic cosine function, cosh x, is defined as (eˣ + e⁻ˣ)/2 and is an even, smooth function. Its Taylor series at 0 includes only even powers of x with positive coefficients, which helps in forming the series for cosh(3x) by substituting 3x into the expansion.