Chapter 8: Q. 59 (page 680)

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .
$\sqrt[3]{1 + x}$

Short Answer

Expert verified

The maclaurin series for the given function is

$\sqrt[3]{1 + x} = 1 + \frac{1}{3} x - \frac{1}{9} x^{2} + \frac{5}{81} x^{3} + \dots$

Step by step solution

Step 1.Given information

We have been given

$f (x) = \sqrt[3]{1 + x}$

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

Step 3. Binomial series for the given function is

so for the given function $f (x) = \sqrt[3]{1 + x}$ , the binomial series is $\sqrt[3]{1 + x} = \sum_{k = 0}^{\infty} (\begin{array}{l} \frac{1}{3} \\ k \end{array}) x^{k}$

this implies that ,

\begin{aligned} \sqrt[3]{1 + x} = (\begin{matrix} \frac{1}{3} \\ 0 \end{matrix}) x^{0} + (\begin{matrix} \frac{1}{3} \\ 1 \end{matrix}) x^{1} + (\begin{matrix} \frac{1}{3} \\ 2 \end{matrix}) x^{2} + (\begin{matrix} \frac{1}{3} \\ 3 \end{matrix}) x^{3} + \dots \\ = 1 + \frac{1}{3} x + \frac{\frac{1}{3} (- \frac{2}{3})}{2!} x^{2} + \frac{\frac{1}{3} (- \frac{2}{3}) (- \frac{5}{3})}{3!} x^{3} + \dots \\ = 1 + \frac{1}{3} x - \frac{1}{9} x^{2} + \frac{5}{81} x^{3} + \dots \end{aligned}

Step 4. Finding the maclaurin series of given function

Hence the maclaurin series of given function is

\sqrt[3]{1 + x} = 1 + \frac{1}{3} x - \frac{1}{9} x^{2} + \frac{5}{81} x^{3} + \dots

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In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .
$\sqrt[3]{1 + x}$

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. Finding the maclaurin series of given function

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In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .1+x3

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. Finding the maclaurin series of given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Theoretical and Mathematical Physics

Pure Maths

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .
$\sqrt[3]{1 + x}$