Question

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$. Note: These are the same functions and values as in Exercises 41–48.

$55.\cos 2x,\frac{\pi }{4}$

Short answer

The Taylor series for the function $f\left(x\right)=\cos \left(2x\right)$at$x=\frac{\mathrm{\pi }}{4}$is$P_n\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}{2}^{2k+1}}{(2k+1)!}{(x-\frac{\pi }{4})}^{2k+1}$

Step-by-step solution

Step 1. Given data

We have the function$f\left(x\right)=\cos \left(2x\right)$

Step 2. Table of the taylor series  

Any function $f$with a derivative of order $n$, the taylor series at $x=\frac{\mathrm{\pi }}{4}$is given by,

$\begin{array}{rl}{P}_n\left(x\right)=& f\left(\frac{\pi }{4}\right)+f^{\mathrm{\prime }}\left(\frac{\pi }{4}\right)(x-\frac{\pi }{4})+\frac{f^{\prime \prime }\left(\frac{\pi }{4}\right)}{2!}{(x-\frac{\pi }{4})}^2+\frac{f^{\prime \prime }\left(\frac{\pi }{4}\right)}{3!}{(x-\frac{\pi }{4})}^3+\frac{f^{\prime \prime \prime \prime }\left(\frac{\pi }{4}\right)}{4!}{(x-\frac{\pi }{4})}^4+\dots \end{array}$

we can write the general of the Taylor series of the function $f$is,

$P_n\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{f^k\left(x_0\right)}{k!}{(x-x_0)}^n$

So, let us first construct the table of the Taylor series for the function $f\left(x\right)=\cos \left(2x\right)$at $x=\frac{\mathrm{\pi }}{4}$

$n$	$f^n\left(x\right)$	$f^n\left(\frac{\mathrm{\pi }}{4}\right)$	$\frac{f^n\left(\frac{\mathrm{\pi }}{4}\right)}{n!}$
0	$\cos 2x$	0	0
1	$-2\sin 2x$	-2	-2
2	$-2^2\cos 2x$	0	0
3	$2^3\sin 2x$	$2^3$	$\frac{2^3}{3!}$
. . .	. . .	. . .	. . .
2k	$(-1{)}^k{2}^{2k}\cos 2x$	0	0
2k+1	$(-1{)}^{k+1}{2}^{2k+1}\sin 2x$	$(-1{)}^{k+1}{2}^{2k+1}$	$\frac{(-1{)}^{k+1}{2}^{2k+1}}{(2k+1)!}$
. . .	. . .	. . .	. . .

Step 3. Taylor series for the f(x)=cos2x

The Taylor series for the function $f\left(x\right)=\cos \left(2x\right)$at $x=\frac{\mathrm{\pi }}{4}$is

$\begin{array}{rl}{P}_n\left(x\right)=& 0+-2\cdot (x-\frac{\pi }{4})+\frac{0}{2!}{(x-\frac{\pi }{4})}^2+\frac{2^3}{3!}{(x-\frac{\pi }{4})}^3+\cdots +\frac{0}{\left(2k\right)!}{(x-\frac{\pi }{4})}^{2k}+\frac{(-1{)}^{k+1}{2}^{2k+1}}{(2k+1)!}{(x-\frac{\pi }{4})}^{2k+1}+\cdots \end{array}$

Or we can write this as$P_n\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}{2}^{2k+1}}{(2k+1)!}{(x-\frac{\pi }{4})}^{2k+1}$