Chapter 8: Q 60. (page 680)

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .
$\frac{1}{(1 + x)^{2}}$

Short Answer

Expert verified

The maclaurin series for the given function is

(1 + x)^{- 2} = 1 - 2 x + 3 x^{2} - 4 x^{3} + \dots

Step by step solution

Step 1. Given information

We have been given

$f (x) = \frac{1}{(1 + x)^{2}}$

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

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

Step 3. Binomial series for the given function is

So for the function $f (x) = \frac{1}{(1 + x)^{2}}$ , the binomial series is

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

implies that ,

\begin{aligned} (1 + x)^{- 2} = (\begin{matrix} - 2 \\ 0 \end{matrix}) x^{0} + (\begin{matrix} - 2 \\ 1 \end{matrix}) x^{1} + (\begin{matrix} - 2 \\ 2 \end{matrix}) x^{2} + (\begin{matrix} - 2 \\ 3 \end{matrix}) x^{3} + \dots \\ = 1 - 2 x + \frac{- 2 (- 3)}{2!} x^{2} + \frac{- 2 (- 3) (- 4)}{3!} x^{3} + \dots \\ = 1 - 2 x + 3 x^{2} - 4 x^{3} + \dots \end{aligned}

Step 4. The maclaurin series for given function is

The maclaurin series of given function $f (x) = \frac{1}{(1 + x)^{2}}$ is

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .
$\frac{1}{(1 + x)^{2}}$

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

Pure Maths

Mechanics Maths

Probability and Statistics

Applied Mathematics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .1(1+x)2

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

Pure Maths

Mechanics Maths

Probability and Statistics

Applied Mathematics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .
$\frac{1}{(1 + x)^{2}}$