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Chapter 10: Problem 12

Use power series operations to find the Taylor series at \(x=0\) for the functions. $$x^{2} \sin x$$

Short Answer

Expert verified
The Taylor series for \( x^2 \sin x \) is \( x^3 - \frac{1}{6}x^5 + \frac{1}{120}x^7 + \cdots \).

Step by step solution

01

Recall the Taylor Series for Sine Function

The Taylor series for the sine function around zero is given by the formula: \( \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1} \). This series expansion will be useful in finding the Taylor series for \( x^2 \sin x \).
02

Multiply the Sine Series by \(x^2\)

The function we are interested in is \( x^2 \sin x \). We multiply each term in the series expansion for \( \sin x \) by \( x^2 \). This gives us: \( x^2 \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+3} \).
03

Write Terms of the New Series

Start substituting the first few values of \( n \) into the series to develop the Taylor series:- For \( n = 0 \): \( \frac{(-1)^0}{1!} x^{3} = x^3 \)- For \( n = 1 \): \( \frac{(-1)^1}{3!} x^{5} = -\frac{1}{6} x^5 \)- For \( n = 2 \): \( \frac{(-1)^2}{5!} x^{7} = \frac{1}{120} x^7 \)Continuing this pattern, the terms start to emerge.
04

Construct the Taylor Series

Based on the terms from Step 3, the Taylor series up to the first three non-zero terms for \( x^2 \sin x \) is: \[ x^2 \sin x = x^3 - \frac{1}{6} x^5 + \frac{1}{120} x^7 + \cdots \].

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Key Concepts

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

Power Series
A power series is a special type of infinite series that is expressed in the form:
\[ \sum_{n=0}^{\infty} a_{n} (x-c)^n \]
Here, \( a_n \) are coefficients, \( x \) is a variable, and \( c \) is a constant called the center of the series.
  • If \( c = 0 \), the power series is centered at zero, which is also known as a Maclaurin series.
  • This structure allows us to represent complex functions through simpler polynomial forms.
  • The series converges within a certain radius of convergence, which depends on the specific function and the center \( c \).
Power series become extremely handy when analyzing and approximating functions in calculus, as they serve as the foundation for Taylor and Maclaurin series expansions.
Sine Function
The sine function, denoted as \( \sin x \), is one of the fundamental trigonometric functions. Its values are derived from the ratios of sides in a right triangle or from unit circle coordinates. The sine function has a distinctive wave-like pattern that repeats every \( 2\pi \) radians.
  • Its Maclaurin series (a special case of the Taylor series at \( x = 0 \)) is given by the alternating infinite series:
    \[ \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1} \]
  • This series is characterized by only odd powers of \( x \) because the sine function's derivatives at zero yield alternating positives and negatives.
  • The series perfectly replicates the \( \sin x \) curve when plotted, offering a way to calculate sine values using basic arithmetic operations without a calculator.
Understanding the sine function's series expansion is crucial when dealing with equations like \( x^2 \sin x \), as it allows us to extend these properties to more complex functions.
Series Expansion
Series expansion involves expressing a function as the sum of terms of a series. This approach provides a polynomial approximation to more complex functions, making calculations easier.
  • The Taylor series is a type of series expansion centered at a point \( c \), allowing one to approximate a function using its derivatives at that point. A Taylor series at \( x = 0 \) is called a Maclaurin series, like the sine series.
  • Expanding functions transforms them into a form with powers of \( x \) and easily computable coefficients. For example, multiplying a series by \( x^2 \) as seen in \( x^2 \sin x \) means adjusting the powers of \( x \) without changing the alternating pattern of coefficients.
  • Recognizing patterns, such as alternating signs or incremental changes in powers and coefficients, simplifies creating a series expansion of any function.
Overall, series expansions are instrumental in analytical mathematics, providing detailed insight into the behavior of functions over an interval of interest.

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Most popular questions from this chapter

a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals. Step 1: Plot the function over the specified interval. Step 2: Find the Taylor polynomials \(P_{1}(x), P_{2}(x),\) and \(P_{3}(x)\) at \(\bar{x}=0\) Step 3: Calculate the ( \(n+1\) )st derivative \(f^{(n+1)}(c)\) associated with the remainder term for each Taylor polynomial. Plot the derivative as a function of \(c\) over the specified interval and estimate its maximum absolute value, \(M\) Step 4: Calculate the remainder \(R_{n}(x)\) for each polynomial. Using the estimate \(M\) from Step 3 in place of \(f^{(n+1)}(c),\) plot \(R_{n}(x)\) over the specified interval. Then estimate the values of \(x\) that answer question (a). Step 5: Compare your estimated error with the actual error \(E_{n}(x)=\left|f(x)-P_{n}(x)\right|\) by plotting \(E_{n}(x)\) over the specified interval. This will help answer question (b). Step 6: Graph the function and its three Taylor approximations together. Discuss the graphs in relation to the information discovered in Steps 4 and 5. $$f(x)=e^{-x} \cos 2 x, \quad|x| \leq 1$$

Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals. Step 1: Plot the function over the specified interval. Step 2: Find the Taylor polynomials \(P_{1}(x), P_{2}(x),\) and \(P_{3}(x)\) at \(\bar{x}=0\) Step 3: Calculate the ( \(n+1\) )st derivative \(f^{(n+1)}(c)\) associated with the remainder term for each Taylor polynomial. Plot the derivative as a function of \(c\) over the specified interval and estimate its maximum absolute value, \(M\) Step 4: Calculate the remainder \(R_{n}(x)\) for each polynomial. Using the estimate \(M\) from Step 3 in place of \(f^{(n+1)}(c),\) plot \(R_{n}(x)\) over the specified interval. Then estimate the values of \(x\) that answer question (a). Step 5: Compare your estimated error with the actual error \(E_{n}(x)=\left|f(x)-P_{n}(x)\right|\) by plotting \(E_{n}(x)\) over the specified interval. This will help answer question (b). Step 6: Graph the function and its three Taylor approximations together. Discuss the graphs in relation to the information discovered in Steps 4 and 5. $$f(x)=(\cos x)(\sin 2 x), \quad|x| \leq 2$$

a. Series for \(\sinh ^{-1} x \quad\) Find the first four nonzero terms of the Taylor series for $$ \sinh ^{-1} x=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}} $$ b. Use the first three terms of the series in part (a) to estimate \(\sinh ^{-1} 0.25 .\) Give an upper bound for the magnitude of the estimation error.

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) \(a_{n}=(0.9999)^{n}\)

The English mathematician Wallis discovered the formula $$\frac{\pi}{4}=\frac{2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdot 8 \cdot \cdots}{3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdot 7 \cdot \cdots}$$ Find \(\pi\) to two decimal places with this formula.
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