Chapter 8: Q 16 (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. $\sin x, x_{0} = \frac{π}{2}$

Short Answer

Expert verified

The Taylor series for the function is $P_{n} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{(2 k)!} {(x - \frac{π}{2})}^{2 k}$

Step by step solution

Given information

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

Find the general of the Taylor series of the function

The Taylor series at $x = \frac{π}{2}$ for any function $f$ with a derivative of order $n$ is given by

$P_{n} (x) = f (\frac{π}{2}) + f^{'} (\frac{π}{2}) (x - \frac{π}{2}) + \frac{f^{''} (\frac{π}{2})}{2!} {(x - \frac{π}{2})}^{2} + f$ ^'''(π2)3!(x-π2)3+f^''''(π2)4!(x-π2)4+....

As a result, first, determine the function's value as well as $f^{'} (x), f^{''} (x), f^{'''} (x)$ at $x = \frac{π}{2}$

Furthermore, the function's general Taylor series is $P_{n} (x) = \sum_{k = 0}^{\infty} \frac{f^{k} (x_{0})}{k!} {(x - x_{0})}^{n}$

Make a table of the Taylor series for the function f(x)=sinx at x=π2

$n$	$f^{n} (x)$	$f^{n} (\frac{π}{2})$	$\frac{f^{n} (\frac{π}{2})}{n!}$
$0$	$\sin x$	$1$	$1$
$1$	$\cos x$	$0$	$0$
$2$	$- \sin x$	$- 1$	$- \frac{1}{2!}$
$3$	$- \cos x$	$0$	$0$
$. . .$	$. . .$	$. . .$	$. . .$
role="math" localid="1653433262354" $2 k$	$(- 1)^{k} \sin x$	$(- 1)^{k}$	$(- 1)^{k} \frac{1}{(2 k)!}$
$2 k + 1$	$(- 1)^{k} \cos x$	$0$	$0$

Find the Taylor series for the function f(x)=sinx at x=π2

The Taylor series for the function $f (x) = \sin x$ at $x = \frac{π}{2}$ is:

$P_{n} (x) = 1 + 0 \cdot (x - \frac{π}{2}) - \frac{1}{2!} {(x - \frac{π}{2})}^{2} + 0 {(x - \frac{π}{2})}^{3} + \dots + \frac{(- 1)^{k}}{(2 k)!} {(x - \frac{π}{2})}^{2 k} + 0 {(x - \frac{π}{2})}^{2 k + 1} + \dots$

Or, we can write as:

$P_{n} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{(2 k)!} {(x - \frac{π}{2})}^{2 k}$

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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. $\sin x, x_{0} = \frac{π}{2}$

Short Answer

Step by step solution

Given information

Find the general of the Taylor series of the function

Make a table of the Taylor series for the function f(x)=sinx at x=π2

Find the Taylor series for the function f(x)=sinx at x=π2

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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=π2

Short Answer

Step by step solution

Given information

Find the general of the Taylor series of the function

Make a table of the Taylor series for the function f(x)=sinx at x=π2

Find the Taylor series for the function f(x)=sinx at x=π2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Statistics

Pure Maths

Theoretical and Mathematical Physics

Applied Mathematics

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. $\sin x, x_{0} = \frac{π}{2}$