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Chapter 8: Q. 61 (page 680)

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .
$\frac{1}{\sqrt{1 + x}}$

Short Answer

Expert verified

The maclaurin series for the given function is

$(1 + x)^{鈭� \frac{1}{2}} = 1 鈭� \frac{1}{2} x + \frac{3}{8} x^{2} 鈭� \frac{5}{16} x^{3} + 鈰�$

Step by step solution

01

Step 1.Given information

We have been given

$\frac{1}{\sqrt{1 + x}}$

to find the maclaurin series by using binomial series

02

Step 2.Defining the series

For any non- zero constant p, the Maclaurin series for the function $g (x) = (1 + x)^{p}$ is called the binomial series which is given by

${鈭�}_{k = 0}^{鈭�} 鈥� (\begin{array}{l} p \\ k \end{array}) x^{k}$

where the binomial coefficient is

$(\begin{array}{l} p \\ k \end{array}) = \{\begin{array}{rr} \frac{p (p 鈭� 1) (p 鈭� 2) 鈰� (p 鈭� k + 1)}{k!}, 鈥呪赌呪赌呪赌� if k > 0 \\ 1, 鈥呪赌呪赌呪赌� if k = 0 \end{array}$

03

Step 3. Binomial series for the given function is

So for the given function $f (x) = \frac{1}{\sqrt{1 + x}}$ binomial series is ,

(1 + x)^{鈭� \frac{1}{2}} = {鈭�}_{k = 0}^{鈭�} 鈥� (\begin{array}{l} 鈭� \frac{1}{2}) x^{k} \\ k \end{array})

implies that ,

\begin{aligned} (1 + x)^{鈭� \frac{1}{2}} = (\begin{array}{c} 鈭� \frac{1}{2} \\ 0 \end{array}) x^{0} + (\begin{array}{c} 鈭� \frac{1}{2} \\ 1 \end{array}) x^{1} + (鈭� \frac{1}{2}) x^{2} + (\begin{array}{c} 鈭� \frac{1}{2}) x^{3} + 鈰� \\ 2 \end{array}) \\ = 1 鈭� \frac{1}{2} x + \frac{鈭� \frac{1}{2} (鈭� \frac{3}{2})}{2!} x^{2} + \frac{鈭� \frac{1}{2} (鈭� \frac{3}{2}) (鈭� \frac{5}{2})}{3!} x^{3} + 鈰� \\ = 1 鈭� \frac{1}{2} x + \frac{3}{8} x^{2} 鈭� \frac{5}{16} x^{3} + 鈰� \end{aligned}

04

Step 4. The maclaurin series for given function is

The maclaurin series for the given function is $(1 + x)^{鈭� \frac{1}{2}} = 1 鈭� \frac{1}{2} x + \frac{3}{8} x^{2} 鈭� \frac{5}{16} x^{3} + 鈰�$

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Most popular questions from this chapter

Find the interval of convergence for power series: ${鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}} {(x + 3)}^{k}$

If f(x) is an nth-degree polynomial and $P_{n} (x)$ is the nth Taylor polynomial for fat $x_{0}$ , what is the nth remainder $R_{n} (x)$ ? What is $R_{n + 1} (x)$ ?

Show that ${鈭�}_{k = 0}^{鈭�} 鈥� \frac{1}{1 + 2^{k}} x^{k}$ , the power series in $x$ from Example 1, diverges when $x = - 2$

What is a Taylor polynomial for a function f at a point $x_{0}$ ?

Use an appropriate Maclaurin series to find the values of the series in Exercises 17鈥�22.

${鈭�}_{k = 0}^{鈭�} {(- \frac{1}{蟺})}^{k}$

See all solutions

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