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

$(1 + x)^{- \frac{1}{2}} = 1 - \frac{1}{2} x + \frac{3}{8} x^{2} - \frac{5}{16} x^{3} + \dots$

Step by step solution

Step 1.Given information

We have been given

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

to find the maclaurin series by using binomial series

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

$\sum_{k = 0}^{\infty} (\begin{array}{l} p \\ k \end{array}) x^{k}$

where the binomial coefficient is

$(\begin{array}{l} p \\ k \end{array}) = \{\begin{array}{rr} \frac{p (p - 1) (p - 2) \dots (p - k + 1)}{k!}, if k > 0 \\ 1, if k = 0 \end{array}$

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)^{- \frac{1}{2}} = \sum_{k = 0}^{\infty} (\begin{array}{l} - \frac{1}{2}) x^{k} \\ k \end{array})

implies that ,

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

Step 4. The maclaurin series for given function is

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

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

Short Answer

Step by step solution

Step 1.Given information

Step 2.Defining the series

Step 3. Binomial series for the given function is

Step 4. The maclaurin series for given function is

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In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .11+x

Short Answer

Step by step solution

Step 1.Given information

Step 2.Defining the series

Step 3. Binomial series for the given function is

Step 4. The maclaurin series for given function is

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Geometry

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

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