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Magazine Chapter 10: Problem 13
Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\frac{x^{2}}{2}-1+\cos x$$
Short Answer
Expert verified
The Taylor series at \(x=0\) for the function is \(\frac{x^4}{24} - \frac{x^6}{720} + \cdots\)."
Step by step solution
01
Identify the Known Taylor Series
We know the Taylor series for the cosine function centered at \(x = 0\) is given by \(\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\). This series will be used directly as part of the solution.
02
Separate the Function Components
Our function is \(\frac{x^2}{2} - 1 + \cos x\). We will deal with each part of this function separately: \(\frac{x^2}{2}\), \(-1\), and \(\cos x\).
03
Write Each Component as a Series
The expression \(\frac{x^2}{2}\) can be written as a power series \(\frac{x^2}{2} = \frac{1}{2}x^2\cdot1\). The constant \(-1\) remains as \(-1\). The Taylor series for \(\cos x\) is \(1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots\).
04
Combine the Series
Start with combining the series for each part: \(\frac{x^2}{2} = 0 + \frac{1}{2}x^2 + 0x^4 + 0x^6 + \cdots\), \(-1 = -1\), and the known series for \(\cos x\). Add them term-by-term: \[1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots\] + \[-1\] + \[0 + \frac{1}{2}x^2 + 0x^4 + \cdots\].
05
Simplify the Combined Series
Cancel or combine coefficients for the powers of \(x\). We have: constant term: \(1 - 1 = 0\), \(x^2\) term: \(-\frac{1}{2}x^2 + \frac{1}{2}x^2 = 0\), the \(x^4\) term remains \(\frac{x^4}{24}\), and higher-order terms follow similarly without contributions from \(\frac{x^2}{2} or -1\).
06
Write Final Taylor Series
The Taylor series at \(x = 0\) for the function \(\frac{x^2}{2} - 1 + \cos x\) is simply \(\frac{x^4}{24} - \frac{x^6}{720} + \cdots\).
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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 way to represent a function as an infinite sum of terms by using powers of a variable, say, \(x\). Think of it as a long polynomial that can go on indefinitely. Power series are incredibly useful because they allow us to express complex functions in simpler terms.
They are written in the form:\[\sum_{n=0}^{finity} a_n (x-c)^n\]Here, \(a_n\) represents the coefficients of the series, and \(c\) is the center of the series. When \(c=0\), we refer to this as a Maclaurin series, a specific kind of Taylor series.
They are written in the form:\[\sum_{n=0}^{finity} a_n (x-c)^n\]Here, \(a_n\) represents the coefficients of the series, and \(c\) is the center of the series. When \(c=0\), we refer to this as a Maclaurin series, a specific kind of Taylor series.
- Power series can approximate functions to a very high degree of accuracy.
- They make it easier to perform calculus operations like integration and differentiation.
- Understanding how to manipulate power series terms is crucial for solving problems like finding Taylor or Maclaurin series.
Cosine Function
The cosine function, known from trigonometry, is more than just a way to determine the adjacent component of an angle in a right triangle. It is also an essential function in calculus and series representation.
In the context of power series, specifically Taylor and Maclaurin series, the cosine function has a well-defined expression:\[\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\]Essentially, the cosine function is represented by an alternating series of even powers of \(x\).
In the context of power series, specifically Taylor and Maclaurin series, the cosine function has a well-defined expression:\[\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\]Essentially, the cosine function is represented by an alternating series of even powers of \(x\).
- The cosine power series converges for all \(x\), making it reliable for many applications.
- It starts with \(1\) and alternates between subtracting and adding terms based on the power of \(x\).
- Understanding the cosine series is crucial when you need to integrate it into more complex functions, as demonstrated in the exercise.
Maclaurin Series
The Maclaurin series is a special case of the Taylor series where the expansion point is at \(x=0\). This makes calculations somewhat simpler, especially when dealing with symmetric functions like \(\cos x\).
Maclaurin series have the general form:\[\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n\]For functions like cosine, this series is quite efficient since many higher-order derivatives are zero or alternate in a predictable pattern.
Maclaurin series have the general form:\[\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n\]For functions like cosine, this series is quite efficient since many higher-order derivatives are zero or alternate in a predictable pattern.
- Maclaurin series help simplify and approximate functions around \(x=0\).
- This form is particularly useful when direct function evaluation is complex or impossible.
- Used widely in mathematics, engineering, and physics, Maclaurin series simplify real-world problem solving.
