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Chapter 9: Problem 4

Which of the series in Exercises are power series? $$x^{7}+x+2$$

Short Answer

Expert verified
The series \(x^7 + x + 2\) is not a power series.

Step by step solution

01

Recognizing a Power Series

A power series is an infinite series of the form \[ \sum_{n=0}^{\infty} c_n (x-a)^n \]where \(c_n\) are coefficients, \(x\) is the variable, and \(a\) is the center of the series.Power series have an infinite number of terms.
02

Analyzing the Given Series

The given series is \(x^7 + x + 2\). This expression has three terms and is not infinite.The terms are \(x^7\), \(x\), and \(2\), which make it a polynomial equation rather than a power series.
03

Conclusion

The expression \(x^7 + x + 2\) is a finite polynomial, not an infinite series.Therefore, it does not satisfy the definition of a power series.

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Key Concepts

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

Polynomial
A polynomial is an expression that consists of finite sums and is key to understanding many mathematical concepts. Consider the expression \(x^7 + x + 2\). This is an example of a polynomial because it contains a finite number of terms:
  • \(x^7\) 鈥� a term of a single variable raised to the 7th power
  • \(x\) 鈥� a term representing the variable itself
  • 2 鈥� a constant term
Polynomials are significant in mathematics because they are relatively simple equations that can model a wide range of real-world phenomena.
Polynomials have a specific structure where the exponents are whole numbers, and the coefficients are constants.
They are an essential foundation for more advanced topics, including power series.
Infinite Series
An infinite series is a sum of infinitely many terms. Unlike polynomials, which are finite, infinite series go on forever.
A power series is a type of infinite series that can be written in the form of \( \sum_{n=0}^{\infty} c_n (x-a)^n \).
Here:
  • \(c_n\) are the coefficients
  • \(x\) is the variable
  • \(a\) is the center of the series
The beauty of infinite series lies in their ability to represent functions with seemingly endless precision.
Infinite series are foundational for calculus, particularly when discussing convergence and power series representation of functions.
Coefficients
Coefficients are the numerical or constant factors in terms of polynomials or series. They multiply the variable's terms.
For example, in the term \(5x^3\), the coefficient is 5. In a power series \( \sum_{n=0}^{\infty} c_n (x-a)^n \), each \(c_n\) is a coefficient.
Coefficients are crucial because:
  • They scale the strength or contribution of each term in the expression
  • They impact the shape and behavior of the function or series graph
Understanding coefficients can help us determine the behavior and properties of a polynomial or series, such as the roots of a polynomial or the convergence of a series.
Convergence
Convergence in the context of series refers to the behavior of the series as the number of terms tends toward infinity.
A series converges if the sum of its terms approaches a specific finite number as more and more terms are added.
With power series, we often discuss the radius of convergence, where:
  • Within this radius, the series converges
  • Outside this radius, the series diverges
Convergence is a critical concept because it establishes the conditions under which infinite series can meaningfully represent functions.
Understanding convergence allows mathematicians to use series in practical applications, ensuring results remain accurate and predictable.

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Most popular questions from this chapter

In \(2013,\) the quantity of copper mined worldwide was 17.9 million tonnes and had been increasing by \(0.025 \%\) annually. By the end of that year, the world's known copper reserves were 690 million tonnes. \(^{4}\) (a) Write and sum a series giving the total quantity of copper mined in the \(n\) years since \(2013 .\) Assume the quantity mined each year continues to increase at \(0.025 \%\) per year, with the first increase in 2014 (b) In what year are the reserves exhausted? (c) How does the sum constructed in part (a) relate to the integral \(\int_{0}^{n} 17.9(1.025)^{t} d t ?\)

Decide if the statements are true or false. Give an explanation for your answer. If \(\sum a_{n}\) is conditionally convergent, then it is absolutely convergent.

If \(a_{n}>0.5 b_{n}>0\) for all \(n\) and \(\sum b_{n}\) diverges, then \(\sum a_{n}\) diverges.

A ball is dropped from a height of 10 feet and bounces. Each bounce is \(\frac{3}{4}\) of the height of the bounce before. Thus, after the ball hits the floor for the first time, the ball rises to a height of \(10\left(\frac{3}{4}\right)=7.5\) feet, and after it hits the floor for the second time, it rises to a height of \(7.5\left(\frac{3}{4}\right)=10\left(\frac{3}{4}\right)^{2}=5.625\) feet. (Assume that there is no air resistance.) (a) Find an expression for the height to which the ball rises after it hits the floor for the \(n^{\text {th }}\) time. (b) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the first, second, third, and fourth times. (c) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the \(n^{\text {th }}\) time. Express your answer in closed form.

The series converges. Is the sum affected by rearranging the terms of the series? $$\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}}$$
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