Q 16 - Calculus

Question

Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.  $\sin x,x_0=\frac{\mathrm{\pi }}{2}$

Step-by-Step Solution

Answer

The Taylor series for the function is $P_n\left(x\right)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{\left(2k\right)!}{\left(x-\frac{\pi }{2}\right)}^{2k}$

Step 1: Given information

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

Step 2: Find the general of the Taylor series of the function

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

$P_n\left(x\right)=f\left(\frac{\mathrm{\pi }}{2}\right)+f^{\text{'}}\left(\frac{\mathrm{\pi }}{2}\right)(x-\frac{\mathrm{\pi }}{2})+\frac{f^{\text{'}\text{'}}\left({\displaystyle \frac{\mathrm{\pi }}{2}}\right)}{2!}{(x-\frac{\mathrm{\pi }}{2})}^2+\frac{f\text{'}\text{'}\text{'}\left(\frac{\mathrm{\pi }}{2}\right)}{3!}{(x-\frac{\mathrm{\pi }}{2})}^3+\frac{f\text{'}\text{'}\text{'}\text{'}\left(\frac{\mathrm{\pi }}{2}\right)}{4!}{(x-\frac{\mathrm{\pi }}{2})}^4+....$

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

Furthermore, the function's general Taylor series is $P_n\left(x\right)=\sum _{k=0}^{\infty }\frac{f^k\left(x_0\right)}{k!}{\left(x-x_0\right)}^n$

Step 3: Make a table of the Taylor series for the function f ( x ) = sin x   at x = π 2  

$n$	$f^n\left(x\right)$	$f^n\left(\frac{\pi }{2}\right)$	$\frac{f^n\left(\frac{\pi }{2}\right)}{n!}$
$0$	$\sin x$	$1$	$1$
$1$	$\cos x$	$0$	$0$
$2$	$-\sin x$	$-1$	$-\frac{1}{2!}$
$3$	$-\cos x$	$0$	$0$
$$\begin{gathered} . \\ . \\ . \end{gathered}$$	$$\begin{gathered} . \\ . \\ . \end{gathered}$$	$$\begin{gathered} . \\ . \\ . \end{gathered}$$	$$\begin{gathered} . \\ . \\ . \end{gathered}$$
$2k$	$(-1{)}^k\sin x$	$(-1{)}^k$	$(-1{)}^k\frac{1}{\left(2k\right)!}$
$2k+1$	$(-1{)}^k\cos x$	$0$	$0$

Step 4: Find the Taylor series for the function f ( x ) = sin x   at x = π 2  

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

$P_n\left(x\right)=1+0·\left(x-\frac{\pi }{2}\right)-\frac{1}{2!}{\left(x-\frac{\pi }{2}\right)}^2+0{\left(x-\frac{\pi }{2}\right)}^3 +\cdots +\frac{(-1{)}^k}{\left(2k\right)!}{\left(x-\frac{\pi }{2}\right)}^{2k}+0{\left(x-\frac{\pi }{2}\right)}^{2k+1}+\cdots$

Or, we can write as:

$P_n\left(x\right)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{\left(2k\right)!}{\left(x-\frac{\pi }{2}\right)}^{2k}$