Question

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\cos 2 x+2 \sin x$$

Short answer

The first four nonzero terms of the Taylor series centered at 0 for the function $$f(x)=\cos 2x+2 \sin x$$ are $$1 + 2x - 2x^2 - \frac{1}{3}x^3$$, and the radius of convergence of the series is infinite.

Step-by-step solution

Find the derivatives of the function

Let's find the first four derivatives of the given function. Function: $$f(x)=\cos 2x+2 \sin x$$ First derivative: $$f'(x)=-2\sin 2x + 2\cos x$$ Second derivative: $$f''(x)=-4\cos 2x - 2\sin x$$ Third derivative: $$f^{(3)}(x)=8\sin 2x - 2\cos x$$ Fourth derivative: $$f^{(4)}(x)=16\cos 2x + 2\sin x$$

Evaluate the derivatives at the center

Now, we'll evaluate each derivative at the center, which is 0. $$f(0)=\cos(2 \cdot 0)+2 \sin 0=1$$ $$f'(0)=-2\sin(2 \cdot 0)+2\cos 0=2$$ $$f''(0)=-4\cos(2 \cdot 0)-2\sin 0=-4$$ $$f^{(3)}(0)=8\sin(2 \cdot 0)-2\cos 0=-2$$ $$f^{(4)}(0)=16\cos(2 \cdot 0)+2\sin 0=16$$

Substitute the evaluated derivatives into the Taylor series expression

The general Taylor series expression is given by: $$f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$$ Now, we substitute the evaluated derivatives into the expression: $$f(x)=\frac{1}{0!}x^{0}+\frac{2}{1!}x+\frac{-4}{2!}x^{2}+\frac{-2}{3!}x^{3}+\frac{16}{4!}x^{4} + \cdots$$

Sum up the first four nonzero terms

To find the first four nonzero terms, we'll sum up the results obtained in the previous step: $$f(x) \approx 1 + 2x - 2x^2 - \frac{1}{3}x^3$$

Determine the radius of convergence

Since the given function contains a sine and cosine function, which have an infinite radius of convergence, the Taylor series representation of the function will also have an infinite radius of convergence. Radius of Convergence: \(\infty\) The first four nonzero terms of the Taylor series centered at 0 for the function $$f(x)=\cos 2x+2 \sin x$$ are $$1 + 2x - 2x^2 - \frac{1}{3}x^3$$, and the radius of convergence of the series is infinite.

Key concepts

Radius of Convergence

The radius of convergence is an important concept when dealing with Taylor series. It indicates the interval over which the series converges to the actual function. In simpler terms, it tells us how far we can go from the center of the series — usually 0 if it's a Maclaurin series — before the series stops accurately representing the function.
For many functions, especially those involving polynomials and rational expressions, the radius of convergence is finite. However, when it comes to trigonometric functions like sine and cosine, they are periodic and inherently well-behaved. As a result, their Taylor series have an infinite radius of convergence.
To find the radius of convergence, one can use the ratio test or the root test. For trigonometric functions, such as in the given problem where you have \(\cos(2x)\) and \(2\sin(x)\), these kinds of series tend to converge over all real numbers, hence an infinite radius of convergence. This characteristic is due to the nature of sine and cosine functions being defined and bounded for all real numbers.

Trigonometric Functions

Trigonometric functions such as sine and cosine are crucial in analyzing periodic phenomena. They are also foundational in calculus and especially in the context of Taylor and Maclaurin series.

• Sine Function: The sine function, \(\sin(x)\), oscillates between -1 and 1. It is defined for all real values of \(x\) and has a period of \(2\pi\). Used in the Taylor series, it provides infinite series representation of periodic patterns.
• Cosine Function: Similarly, the cosine function, \(\cos(x)\), also oscillates between -1 and 1, with the same periodic behavior. It's often used alongside sine in series expansions due to its complementary properties and phase shift.

To express these functions as Taylor series, you expand them around a point, typically zero, turning them into fun and breathtaking sums of infinite terms. These expansions capture the essence of these functions over their entire domain thanks to their infinite radius of convergence.

Maclaurin Series

A Maclaurin series is a type of Taylor series expanded around zero (0). It is a way to express a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For many functions, the Maclaurin series is not only an approximation but can be exactly equal to the function over its interval of convergence.
For the function \(f(x) = \cos(2x) + 2 \sin(x)\), the Maclaurin series is built by evaluating the function and its derivatives at zero and then representing it using these values.

• The first term: The function value at \(x=0\).
• Subsequent terms: Derived by taking successive derivatives of the function, evaluating them at zero, and dividing by factorial terms.

The beauty of the Maclaurin series lies in its simplicity in capturing complex behaviors of functions like \(\cos\) and \(\sin\) using simple polynomial expressions. This approach not only simplifies computations but also provides significant insights into the nature of periodic and oscillating functions.