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Chapter 9: Problem 25

Using a Binomial Series In Exercises \(21-26\) use the binomial series to find the Maclaurin series for the function. $$f(x)=\sqrt[4]{1+x}$$

Short Answer

Expert verified
The Maclaurin series representation of the function \(f(x) = \sqrt[4]{1+x}\) is: \(1 + \frac{1}{4}x -\frac{3}{32}x^2 + \frac{1}{64}x^3 -\frac{5}{256}x^4 + ... \).

Step by step solution

01

Express function in form (1+x)^n

Rearranging the function, it can be written as \(f(x) = (1+x)^\frac{1}{4}\), where \(n = \frac{1}{4}\). This fits the form required to use the binomial series.
02

Write down binomial series

The binomial series expansion would be \[\sum_{k=0}^{\infty} {n \choose k} x^k = (1+x)^n\], where \({n \choose k}\) are the binomial coefficients.
03

Apply binomial series

Applying the binomial series for \(n=\frac{1}{4}\), the series becomes \(f(x) = \sum_{k=0}^{\infty} {\frac{1}{4} \choose k} x^k\), where \({\frac{1}{4} \choose k}\) would give the following series : \(1, \frac{1}{4}, -\frac{3}{32}, \frac{1}{64}, -\frac{5}{256}, ...\). These coefficients along with \(x^k\) form the series representation of the function.
04

Write out first few terms

The first few terms of the series are \(1 + \frac{1}{4}x -\frac{3}{32}x^2 + \frac{1}{64}x^3 -\frac{5}{256}x^4 + ...\). The general form of the terms of series can be written as \((...\times(-1)^{k-1}x^k)\), with each successive term alternating in sign and being multiplied by \(x\).

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Key Concepts

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

Binomial Series
The binomial series is a powerful tool in mathematics that allows us to expand expressions of the form \((1 + x)^n\) into an infinite series. This is especially useful when working with functions that can be rewritten or approximated in this form. The key idea is that we are representing a power of a binomial expression using an infinite sum.
  • The binomial series formula is given by \(\sum_{k=0}^{\infty} {n \choose k} x^k = (1+x)^n\).
  • This formula applies as long as \(|x| < 1\), ensuring the series converges.
By using the binomial series, we can represent various functions in a form that is much easier to work with for both analytical and numerical purposes.
Binomial Coefficients
Binomial coefficients are a crucial component of the binomial series and play a significant role in many areas of mathematics. These coefficients \({n \choose k}\), represent the coefficients of the expanded terms in a binomial series.
  • The general formula for a binomial coefficient is \({n \choose k} = \frac{n(n-1)(n-2)...(n-k+1)}{k!}\).
  • They represent the number of ways to choose \(k\) elements from a set of \(n\) elements, hence are called 'coefficients.'
  • When \(n\) is a non-integer, as in our original exercise \(n = \frac{1}{4}\), the coefficients are computed using fractional values, allowing even fractional powers to be expanded.
Understanding and working with binomial coefficients facilitate the calculation of terms in a binomial series, particularly when expanding expressions like \((1+x)^\frac{1}{4}\).
Series Expansion
Series expansion involves expressing a function as an infinite sum of terms. This technique is particularly useful in calculus and numerical analysis, as it simplifies complex functions into more manageable forms.
  • In our problem, the series expansion is based on the binomial series and uses this concept to simplify the function \(f(x) = \sqrt[4]{1+x}\).
  • The series expansion of a function represents it as a combination of simpler, easily computable terms, each multiplied by a power of \(x\).
  • The initial few terms, such as \(1 + \frac{1}{4}x -\frac{3}{32}x^2\), help in approximating the function for small values of \(x\), providing insights into the behavior of the function close to zero.
Using series expansions, one can approximate the values of functions, aiding in solving equations analytically or implementing them in computational algorithms.
Function Representation
In mathematics, representing a function in its simplest, most useful form is often crucial. For functions like \(f(x) = \sqrt[4]{1+x}\), the representation using a Maclaurin series provides several advantages.
  • A Maclaurin series is essentially a Taylor series centered at 0, presenting the function as an infinite polynomial.
  • This approach allows us not only to approximate the function for small values of \(x\) but also to gain a deeper understanding of its properties, such as continuity and differentiability.
  • The expansion using the binomial series gives us a practical way to compute function values, derivatives, or integrals.
Function representation using series expansion transforms complex algebraic expressions into an infinite sequence of terms, making them easier to analyze or use in further calculations.

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Most popular questions from this chapter

$$\begin{array}{l}{\text { Proof Prove that the power series }} \\\ {\sum_{n=0}^{\infty} \frac{(n+p) !}{n !(n+q) !} x^{n}} \\ {\text { has a radius of convergence of } R=\infty \text { when } p \text { and } q \text { are }} \\ {\text { positive integers. }}\end{array}$$

Finding Intervals of Convergence In Exercises \(49-52\) , find the intervals of convergence of $$f(x),\left(\text { b) } f^{\prime}(x),\left(\text { c) } f^{\prime \prime}(x), \text { and }\left(\text { d) } \int f(x) d x\right.\right.\right.$$ (Be sure to include a check for convergence at the endpoints of the intervals.) $$f(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-5)^{n}}{n 5^{n}}$$

$$\begin{array}{l}{\text { Differential Equation In Exercises } 59-64, \text { show that the }} \\ {\text { function represented by the power series is a solution of the }} \\ {\text { differential equation. }}\end{array}$$ $$y=\sum_{n=0}^{\infty} \frac{(-1)^{n}+x^{2 n+1}}{(2 n+1) !}, \quad y^{\prime \prime}+y=0$$

Finding Intervals of Convergence In Exercises \(49-52\) , find the intervals of convergence of $$f(x),\left(\text { b) } f^{\prime}(x),\left(\text { c) } f^{\prime \prime}(x), \text { and }\left(\text { d) } \int f(x) d x\right.\right.\right.$$ (Be sure to include a check for convergence at the endpoints of the intervals.) $$f(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-2)^{n}}{n}$$

Area In Exercises 71 and \(72,\) use a power series to approximate the area of the region with an error of less than \(0.0001 .\) Use a graphing utility to verify the result. $$\int_{0.5}^{1} \cos \sqrt{x} d x$$
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