Convergence Write the remainder term Rₙ(x) for the Taylor series for the following functions centered at the given point a. Then show that lim ₙ → ∞ |Rₙ(x)| = 0, for all x in the given interval.
ƒ(x) = sinh x + cosh x, a = 0, - ∞ < x < ∞

Verified step by step guidance

1. Identify the function and the center of the Taylor series expansion. Here, the function is \(f(x) = \sinh x + \cosh x\) and the center is \(a = 0\).
2. Recall the Taylor series remainder term in Lagrange form: \[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x - a)^{n+1}\] where \(c\) is some number between \(a\) and \(x\).
3. Find the \((n+1)\)-th derivative of \(f(x)\). Since \(f(x) = \sinh x + \cosh x\), note that the derivatives cycle in a predictable way. Compute \(f^{(n+1)}(x)\) explicitly or recognize the pattern.
4. Express the remainder term \(R_n(x)\) using the derivative found, substituting \(a=0\): \[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\] with \(c\) between \$0\( and \)x$.
5. To show that \(\lim_{n \to \infty} |R_n(x)| = 0\) for all real \(x\), analyze the growth of the numerator \(|f^{(n+1)}(c)|\) and the factorial in the denominator. Use the fact that factorial growth dominates exponential growth to conclude the limit is zero.

Key Concepts

Taylor Series and Remainder Term

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point. The remainder term Rₙ(x) measures the error between the function and its nth-degree Taylor polynomial. It is often expressed using the Lagrange form, involving the (n+1)th derivative evaluated at some point between a and x.

Hyperbolic Functions and Their Derivatives

The functions sinh x and cosh x are hyperbolic sine and cosine, respectively, with derivatives that cycle predictably: d/dx(sinh x) = cosh x and d/dx(cosh x) = sinh x. Understanding these derivatives helps in finding the Taylor series terms and the remainder for the combined function f(x) = sinh x + cosh x.

Limit of the Remainder Term and Convergence

To prove convergence of the Taylor series, we show that the remainder term Rₙ(x) approaches zero as n approaches infinity for all x in the interval. This involves bounding the remainder term and using properties of the function and its derivatives to demonstrate that the error vanishes, ensuring the series converges to the function.