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Chapter 1: Q44P (page 32)

$f (x) = 鈭� x, a = 25 .$

Short Answer

Expert verified

The first few terms of Taylor鈥檚 series expansion are:

$\sqrt{x} = 5 + (\frac{x - 25}{10}) - \frac{(x - 25)^{2}}{1000} + \frac{(x - 25)^{3}}{50000} + 鈥�$

Step by step solution

01

Given Information

The given function.

$f (x) = 鈭� x$

02

Definition of Taylor’s series

The Taylor鈥檚 series is a power series that yields the expansion of a function in the vicinity of a point if the function is continuous, all of its derivatives exist, and the series converges.

03

Find the Taylor’s series for x

Rewrite $\sqrt{x}$ ,

$\sqrt{x} = \sqrt{25 + (x - 25)} \sqrt{x} = 5 \sqrt{1 + \frac{(x - 25)}{25}}$

Use the Maclaurin series of $\sqrt{1 + x}$ and replace $x$ by $\frac{x - 25}{25}$ ,

localid="1657421695758" $\sqrt{1 + x} = 1 + \frac{x}{2} - \frac{x^{2}}{8} + \frac{x^{3}}{16} - \frac{5 x^{4}}{128} + 鈥� \sqrt{1 + \frac{(x - 25)}{25}} = 1 + \frac{(\frac{x - 25}{25})}{2} - \frac{{(\frac{x - 25}{25})}^{2}}{8} + \frac{{(\frac{x - 25}{25})}^{3}}{16} + 鈥� \sqrt{x} = 5 脳 \sqrt{1 + \frac{(x - 25)}{25}}$

$= 5 + (\frac{x - 25}{10}) - \frac{(x - 25)^{2}}{1000} + \frac{(x - 25)^{3}}{50000} + 鈥�$

Hence, first few terms of Taylor鈥檚 series expansion about $a = 25$ are:

$\sqrt{x} = 5 + (\frac{x - 25}{10}) - \frac{(x - 25)^{2}}{1000} + \frac{(x - 25)^{3}}{50000} + 鈥�$

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Most popular questions from this chapter

Show that if p is a positive integer, then $(\begin{array}{l} p \\ n \end{array}) = 0$ when $n > p$ , so $(1 + x)^{p} = 鈭� (\begin{array}{l} p \\ n \end{array}) x^{n}$ is just a sum of $p + 1$ terms, from $n = 0$ to $n = p$ . For example, $(1 + x)^{2}$ has $3$ terms, $(1 + x)^{3}$ has $4$ terms, etc. This is just the familiar binomial theorem.

${鈭�}_{n = 1}^{鈭�} 鈥� (\sin 鈦� x)^{n} (鈭� 1)^{n} 2^{n}$

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case ${鈭�}_{n = 1}^{鈭�} \frac{x^{2 n}}{2^{n} n^{2}}$

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\lim_{x 鈫� 0} \frac{\sin x - x}{x^{3}}$ .

$\ln \sqrt{\frac{1 + x}{1 - x}} = {鈭�}_{0}^{x} \frac{d t}{1 - t^{2}}$

See all solutions

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