Question

Find the remainder in the Taylor series centered at the point a for the following functions. Then show that \(\lim R_{n}(x)=0\) for all \(x\) in the interval of convergence. $$f(x)=\cos 2 x, a=0$$

Short answer

In this exercise, we found the remainder of the Taylor series centered at 0 for the function \(f(x)=\cos 2x\). We determined the Taylor series expansion of the given function and identified that the remainder term \(R_n(x)\) goes to 0 as the degree of the polynomial increases. This means that the Taylor series expansion converges to the function for all \(x\) in the interval of convergence.

Step-by-step solution

Find the Taylor series expansion of \(f(x)\) at \(a=0\)

To find the Taylor series expansion, we first differentiate the given function repeatedly and evaluate it at \(a=0\). Then substitute the derivatives into the Taylor series expansion formula. For \(f(x) = \cos 2x\), we have: $$f'(x) = -2\sin 2x$$ $$f''(x) = -4\cos 2x$$ $$f'''(x) = 8\sin 2x$$ $$f^{(4)}(x) = 16\cos 2x$$ $$\cdots$$ Now, we evaluate these derivatives at \(a=0\): $$f(0) = \cos 2(0) = 1$$ $$f'(0) = -2\sin 2(0) = 0$$ $$f''(0) = -4\cos 2(0) = -4$$ $$f'''(0) = 8\sin 2(0) = 0$$ $$f^{(4)}(0) = 16\cos 2(0) = 16$$ $$\cdots$$ From these results, we can write the Taylor series expansion of \(f(x)\) at a=0: $$f(x) = 1 - \frac{4}{2!}x^2 + \frac{16}{4!}x^4 - \cdots$$

Identify the remainder term and find its limit

The remainder term, \(R_n(x)\), is given by: $$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-0)^{n+1}$$ where \(c\) is some number between \(0\) and \(x\). Since the cosine and sine functions are periodic, the derivatives of the cosine function will alternate between cosine and sine, whose values are always between \(-1\) and \(1\). Thus, \(|f^{(n+1)}(c)|\leq1\) for any \(c\). We now examine the limit as \(n\) approaches infinity: $$\lim_{n\to\infty} R_n(x) = \lim_{n\to\infty} \frac{|f^{(n+1)}(c)|}{(n+1)!}x^{n+1} \leq \lim_{n\to\infty} \frac{1}{(n+1)!}x^{n+1} = 0$$ In the last step, we used the fact that the limit of any polynomial divided by a factorial goes to zero. This shows that the remainder term goes to zero as the degree of the polynomial increases, meaning that the Taylor series expansion converges to the function for all \(x\) in the interval of convergence.

Key concepts

Remainder Term

In the context of Taylor series, the remainder term, often denoted as \( R_n(x) \), represents the error between the actual function \( f(x) \) and the approximated function given by the Taylor polynomial of degree \( n \). This provides a way to understand how accurately the Taylor polynomial estimates the function. Particularly, the formula for the remainder term is:

• \( R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} \)

Here, \( c \) is a point in the interval \((a, x)\), and \( f^{(n+1)}(c) \) represents the \((n+1)\)-th derivative of \( f \) evaluated at \( c \).

This remainder term is crucial because it helps in analyzing how the Taylor series behaves as more terms are included. As shown, if the limit of the remainder term as \( n \) approaches infinity equals zero, it implies that the series representation converges perfectly to the actual function value, which in this specific example directly demonstrates the precision of the Taylor series.

Therefore, knowing about the remainder term helps to gauge the quality and accuracy of a Taylor series approximation which directly impacts the reliability of using these approximations in practical applications.

Convergence

When discussing Taylor series, convergence is a concept that illustrates whether adding an infinite number of terms in the series results in the approximation becoming the exact value of the function. For a Taylor series centered at point \( a \) for a function \( f(x) \), convergence occurs within an interval around \( a \). This interval is known as the interval of convergence.

The series' convergence is often determined using the remainder term \( R_n(x) \). If \( \lim_{n \to \infty} R_n(x) = 0 \) holds for values within this interval, it signifies that the infinite Taylor series converges to the function itself over that domain. This signifies that the deviation or the "error" made by the approximating function shrinks as the number of terms increases.

• In practical terms, it means the Taylor series provides better approximations as you progress with more terms within its radius.
• For periodic functions like trigonometric functions, the smoother nature enhances convergence behavior over larger intervals.

Understanding convergence is fundamental, particularly for predicting how many terms might be necessary to achieve a desired degree of accuracy in approximations using Taylor series.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are vital components in mathematics, representing periodic phenomena in nature. These functions are both periodic and continuously differentiable, making them excellent candidates for expressing through Taylor series – a key technique in calculus.

• The cosine function, in particular, is infinitely differentiable, with derivatives alternating in sign and nature (cosine and sine), each derivative evaluated at zero resulting in an orderly alternating series.
• Due to their predictable nature, trigonometric functions allow simplification and effective use in approximations with Taylor polynomials.

The Taylor series expansion for \( \cos 2x \) around \( a=0 \) generates a polynomial that mirrors the function’s periodic behavior closely within certain intervals. This expands our ability to analyze and calculate values of the function without direct evaluation at any given point.

By expressing trigonometric functions through Taylor series, engineers and scientists simplify complex calculations in fields like signal processing, mechanical oscillations, and even quantum mechanics, where precise approximations are often essential. Understanding how these functions behave when decomposed into series aids in comprehending broader applications in both pure and applied mathematical contexts.