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Chapter 10: Problem 81

Find the next two terms of the following Taylor series. $$\frac{1}{\sqrt{1+x}} 1-\frac{1}{2} x+\frac{1 \cdot 3}{2 \cdot 4} x^{2}-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} x^{3}+\cdots$$

Short Answer

Expert verified
Answer: The next two terms in the Taylor series for the function \(f(x) = \frac{1}{\sqrt{1+x}}\) are \(\frac{1 \cdot 3 \cdot 5 \cdot 7}{2 \cdot 4 \cdot 6 \cdot 8}x^4\) and \(-\frac{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9}{2 \cdot 4 \cdot 6 \cdot 8 \cdot 10} x^5\).

Step by step solution

01

1. Understanding the given Taylor series

The given series is: $$\frac{1}{\sqrt{1+x}} = 1-\frac{1}{2} x+\frac{1 \cdot 3}{2 \cdot 4} x^{2}-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} x^{3}+\cdots$$ We can see that the coefficients of the Taylor series have alternating signs and a pattern in the terms of the numerators and denominators.
02

2. Finding the general term of the Taylor series

By analyzing the series' coefficients, we can guess the general term of the Taylor series. We note that every term has a numerator with consecutive odd numbers and a denominator with consecutive even numbers. Additionally, we can see that the series has alternating signs. Hence, we can express the nth term of the series as: $$T_n = (-1)^n \cdot \frac{(2n-1)!!}{(2n)!!}x^n$$ Where \(n=0, 1, 2, \cdots\) and \(!!\) denotes the double factorial function (product of integers spaced by two, e.g., \(5!!=5\cdot3\cdot1\)).
03

3. Calculating the next term (Term 4) of the Taylor series

To find the fourth term of the series, we will evaluate the general term when \(n=4\), using the expression we found in step 2: $$T_4=(-1)^4 \cdot \frac{(2 \cdot 4 - 1)!!}{(2 \cdot 4)!!}x^4 = \frac{(7)!!}{(8)!!}x^4 = \frac{1 \cdot 3 \cdot 5 \cdot 7}{2 \cdot 4 \cdot 6 \cdot 8} x^4$$
04

4. Calculating the next term (Term 5) of the Taylor series

To find the fifth term of the series, we will evaluate the general term when \(n=5\), using the expression we found in step 2: $$T_5=(-1)^5 \cdot \frac{(2 \cdot 5 - 1)!!}{(2 \cdot 5)!!}x^5 = -\frac{(9)!!}{(10)!!}x^5 = -\frac{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9}{2 \cdot 4 \cdot 6 \cdot 8 \cdot 10} x^5$$
05

5. Writing the updated Taylor series with the found terms

Now that we have found the next two terms of the Taylor series, we can write the updated series as: $$\frac{1}{\sqrt{1+x}} = 1-\frac{1}{2} x+\frac{1 \cdot 3}{2 \cdot 4} x^{2}-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} x^{3} + \frac{1 \cdot 3 \cdot 5 \cdot 7}{2 \cdot 4 \cdot 6 \cdot 8}x^4 -\frac{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9}{2 \cdot 4 \cdot 6 \cdot 8 \cdot 10} x^5+\cdots$$

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Most popular questions from this chapter

Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. $$f(x)=1 / \sqrt{x} \text { with } a=4 ; \text { approximate } 1 / \sqrt{3}$$

Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). $$\lim _{x \rightarrow 0} \frac{e^{a x}-1}{x}$$

If the power series \(f(x)=\sum c_{k} x^{k}\) has an interval of convergence of \(|x|

Errors in approximations Suppose you approximate \(\sin x\) at the points \(x=-0.2,-0.1,0.0,0.1,\) and 0.2 using the Taylor polynomials \(p_{3}=x-x^{3} / 6\) and \(p_{5}=x-x^{3} / 6+x^{5} / 120 .\) Assume that the exact value of \(\sin x\) is given by a calculator. a. Complete the table showing the absolute errors in the approximations at each point. Show two significant digits. $$\begin{array}{|c|l|l|} \hline x & \text { Error }=\left|\sin x-p_{3}(x)\right| & \text { Error }=\left|\sin x-p_{5}(x)\right| \\ \hline-0.2 & & \\ \hline-0.1 & & \\ \hline 0.0 & & \\ \hline 0.1 & & \\ \hline 0.2 & & \\ \hline \end{array}$$ b. In each error column, how do the errors vary with \(x\) ? For what values of \(x\) are the errors the largest and smallest in magnitude?

By comparing the first four terms, show that the Maclaurin series for \(\cos ^{2} x\) can be found (a) by squaring the Maclaurin series for \(\cos x,\) (b) by using the identity \(\cos ^{2} x=(1+\cos 2 x) / 2,\) or \((\mathrm{c})\) by computing the coefficients using the definition.
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