Chapter 8: Q 15 (page 704)

Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.
$\sin x, x_{0} = 0$

Short Answer

Expert verified

The Maclurin series for the function is $f (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{(2 k + 1)!} x^{2 k + 1}$

Step by step solution

Given information

The function is $f (x) = \sin x$

Find the general of the Maclurin series of the function

The Maclurin series at $x_{0} = 0$ for any function $f$ with a derivative of order $n$ is given by

$f (x) = f (0) + f^{'} (0) x + \frac{f^{''} (0)}{2!} x^{2} + \frac{f^{'''} (0)}{3!} x^{3} + \frac{f^{''''} (0)}{4!} x^{4} + \dots$

The function's general Maclurin series is

f (x) = \sum_{n = 0}^{\infty} \frac{f^{n} (0)}{n!} x^{n}

Make a table of the Maclurin series for the function f(x)=sinx

$n$	$f^{n} (x)$	$f^{n} (0)$	$\frac{f^{n} (0)}{n!}$
$0$	$\sin x$	$0$	$0$
$1$	$\cos x$	$1$	$1$
$2$	$- \sin x$	$0$	$0$
$3$	$- \cos x$	$- 1$	$- \frac{1}{3!}$
$. . .$	$. . .$	$. . .$	$. . .$
$2 k$	$(- 1)^{k} \sin x$	$0$	$0$
$2 k + 1$	$(- 1)^{k} \cos x$	$(- 1)^{k}$	$(- 1)^{k} \frac{1}{(2 k + 1)!}$

Find the Maclurin series for the function f(x)=sinx

The Maclurin series for the function $f (x) = \sin x$ is:

$0 + 1 \cdot x + \frac{0}{2!} x^{2} + \frac{(- 1)}{3!} x^{3} + 0 x^{4} + \frac{1}{5!} x^{5} + \frac{0}{6!} x^{6} + \dots$

Or, we can write as:

$f (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{(2 k + 1)!} x^{2 k + 1}$

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Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.
$\sin x, x_{0} = 0$

Short Answer

Step by step solution

Given information

Find the general of the Maclurin series of the function

Make a table of the Maclurin series for the function f(x)=sinx

Find the Maclurin series for the function f(x)=sinx

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Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.sinx,x0=0

Short Answer

Step by step solution

Given information

Find the general of the Maclurin series of the function

Make a table of the Maclurin series for the function f(x)=sinx

Find the Maclurin series for the function f(x)=sinx

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Applied Mathematics

Geometry

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.
$\sin x, x_{0} = 0$