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Chapter 6: Problem 204

Find the Maclaurin series of each function. $$ f(x)=\frac{\sin x}{x} $$

Short Answer

Expert verified
The Maclaurin series for \( \frac{\sin x}{x} \) is \( 1 - \frac{x^2}{6} + \frac{x^4}{120} - \cdots \).

Step by step solution

01

Understand the Definition of Maclaurin Series

A Maclaurin series is a special case of the Taylor series where the function is expanded at 0. The general form for the Maclaurin series of a function \( f(x) \) is given by \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \)
02

Find the Maclaurin Series for sin(x)

Identify the Maclaurin series expansion for \( \sin x \), which is \( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \).
03

Divide the Series by x

Divide each term of the series \( \sin x \) by \( x \) to find the series for \( \frac{\sin x}{x} \). This results in \( \frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots \).
04

Combine and Simplify

Write the final Maclaurin series for \( \frac{\sin x}{x} \) as \( \frac{\sin x}{x} = 1 - \frac{x^2}{6} + \frac{x^4}{120} - \frac{x^6}{5040} + \cdots \). This expression shows the function's behavior near \( x = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Taylor series
The Taylor series is a powerful tool in mathematics used for function approximation. It allows us to expand a given function into an infinite sum of terms calculated from the function's derivatives at a single point. Essentially, it takes a complex function and represents it as an infinite polynomial of powers of its variable.
  • The general formula for a Taylor series is: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \]
  • This expansion is centered around the point \( a \).
  • The derivatives \( f'(a), f''(a) \) and so on, are evaluated at this specific point.
This makes the Taylor series a very versatile tool as it can be tailored to approximate behavior near any desired point \( a \). It is especially useful for understanding the behavior of functions around a point in calculus, physics, and engineering.
series expansion
Series expansion is a method we use to express functions as sums of simpler components, usually involving powers or exponential terms. Understanding a series can provide insight into the function's behavior over an interval or point.
  • A series expansion breaks functions down into linear combinations of basis functions, like polynomials or trigonometric components.
  • It's useful for finding approximate values for complex functions where direct computation might be difficult.
In the context of calculus and analysis, series expansions are often derived from Taylor or Maclaurin series. A function can be approached through its series to gain insights into characteristics such as convergence, bounds, and approximation errors.
function expansion at zero
In mathematics, expanding a function at zero is specifically known as a Maclaurin series. It is a special case of the Taylor series where the expansion occurs around \( x = 0 \).
  • The Maclaurin series simplifies the general Taylor series formula:\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \]
  • This approach provides a straightforward way to understand a function's behavior very close to zero.
This type of expansion is common for functions that are easy or exactly calculable at zero, as many elementary functions tend to.
This was used in the example of \( f(x)=\frac{\sin x}{x} \), focused on understanding the function's behavior as \( x \) approaches zero, leading to easier analysis and comprehension of key properties.
trigonometric functions
Trigonometric functions, such as sine and cosine, are fundamental components of geometry and calculus. They also play a key role in many analytical and physical applications.
  • They describe the ratios of the sides of a right triangle in terms of its angles, commonly denoted as \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \).
  • These functions are periodic, meaning they repeat values in a regular interval, making them essential for modeling cyclic phenomena such as sound and light waves.
In series expansion, these trigonometric functions can be broken down using Taylor and Maclaurin series. For example, the Maclaurin series of \( \sin x \) is:\[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]By using these series, we can approximate trigonometric functions in computations where analytic solutions are challenging to find, enhancing the understanding and prediction of behaviors for various inputs.

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Most popular questions from this chapter

The exercise make use of the functions \(S_{5}(x)=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}\) and \(C_{4}(x)=1-\frac{x^{2}}{2}+\frac{x^{4}}{24}\) on \([-\pi, \pi]\). (Taylor approximations and root finding.) Recall that Newton's method \(x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}\) approximates solutions of \(f(x)=0\) near the input \(x_{0}\). a. If \(f\) and \(g\) are inverse functions, explain why a solution of \(g(x)=a\) is the value \(f(a)\) of \(f\). b. Let \(p_{N}(x)\) be the \(N\) th degree Maclaurin polynomial of \(e^{x}\). Use Newton's method to approximate solutions of \(p_{N}(x)-2=0\) for \(N=4,5,6\) c. Explain why the approximate roots of \(p_{N}(x)-2=0\) are approximate values of \(\ln (2)\)

Use appropriate substitutions to write down the Maclaurin series for the given binomial. $$ (1-x)^{1 / 3} $$

Use the substitution \((b+x)^{r}=(b+a)^{r}\left(1+\frac{x-a}{b+a}\right)^{r} \quad\) in the binomial expansion to find the Taylor series of each function with the given center. $$ \sqrt{x} \text { at } a=4 $$

Use the fact that if \(q(x)=\sum_{n=1}^{\infty} a_{n}(x-c)^{n} \quad\) converges in an interval containing \(c,\) then \(\lim _{x \rightarrow c} q(x)=a_{0}\) to evaluate each limit using Taylor series. $$ \lim _{x \rightarrow 0^{+}} \frac{\cos (\sqrt{x})-1}{2 x} $$

Use appropriate substitutions to write down the Maclaurin series for the given binomial. $$ (1-2 x)^{2 / 3} $$
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