# Taylor series a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. f(x)=cos x,a=pi

Question
Series
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
$$f(x)=\cos x,a=\pi$$

2020-11-27
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
The value of $$f(x)=\cos x$$ and its derivatives at $$x=\pi$$ are computed in the below table,
$$\begin{array}{|c|c|}\hline k&f^k(x)&f^k(\pi)\\\hline0&f(x)=\cos x&f(\pi)=-1\\\hline1&f'(x)=-\sin x&f'(\pi)=0\\\hline2&f''(x)=-\cos x&f''(\pi)=1\\\hline3&f'''(x)=\sin x&f'''(\pi)=0\\\hline4&f^4(x)=\cos x&f^4(\pi)=-1\\\hline5&f^5(x)=-\sin x&f^5(\pi)=0\\\hline6&f^6(x)=-\cos x&f^6(\pi)=1\\\hline\end{array}$$
The Taylor series centered at $$a=\pi$$ is computed as follows,
$$f(x)=f(\pi)+f'(\pi)(x)+\frac{f''(\pi)(x^2)}{2!}+\frac{f'''(\pi)(x^3}{3!}+...$$
Plug the required values from the table in the above,
$$f(x)=-1+0(x-\pi)+\frac{(1)}{2!}(x-\pi)^2+\frac{0}{3!}(x-\pi)^3+\frac{(-1)}{4!}(x-\pi)^4+\frac{0}{5!}(x-\pi)^5+\frac{(1)}{6!}(x-\pi)^6+...$$
$$=-1+0+\frac{(1)}{2!}(x-\pi)^2+0+\frac{(-1)}{4!}(x-\pi)^4+0+\frac{(1)}{6!}(x-\pi)^6+...$$
$$=-1+\frac12(x-\pi)^2-\frac{1}{24}(x-\pi)^4+\frac{1}{720}(x-\pi)^6+...$$
Therefore, the first four nonzero terms of the Taylor series centered at $$a=\pi$$ is
$$-1+\frac12(x-\pi)^2-\frac{1}{24}(x-\pi^4)+\frac{1}{720}(x-\pi)^6+...$$
b). From part (a), the first four nonzero terms of the Taylor series centered at $$a=\pi$$ is
$$-1+\frac12(x-\pi)^2-\frac{1}{24}(x-\pi^4)+\frac{1}{720}(x-\pi)^6+...$$
Here, the series is alternative and the general kth term can be written as $$\frac{x^{2k}}{(2k)!}(x-\pi)^{2k}$$
$$-1+\frac12(x-\pi)^2-\frac{1}{24}(x-\pi^4)+\frac{1}{720}(x-\pi)^6+...=\sum_{k=0}^\infty\frac{x^{k+1}}{(2k)!}(x-\pi)^{2k}$$
Therefore, the summation notation of the series is $$\sum_{k=0}^\infty\frac{x^{k+1}}{(2k)!}(x-\pi)^{2k}$$

### Relevant Questions

Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
$$f(x)=x\ln x-x+1,a=1$$
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
$$f(x)=2^x,a=1$$
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
c. Determine the interval of convergence of the series.
$$f(x)=\log_3(x+1),a=0$$
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
$$f(x)=\cosh(2x-2),a=1$$
Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
$$f(x)=\tan^{-1}4x,a=0$$
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
$$\displaystyle{f{{\left({x}\right)}}}={\text{cosh}{{\left({2}{x}-{2}\right)}}},{a}={1}$$
Any method
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.
b. Determine the radius of convergence of the series.
$$f(x)=\cos2x+2\sin x$$
$$f(x)=x^2\cos x^2$$
$$f(x)=(1+x)^{\frac{2}{3}}$$, approximate $$(1.02)^{\frac{2}{3}}$$
Use the definition of Taylor series to find the Taylor series, centered at c, for the function. $$f(x)=\cos x,\ c=\frac{-\pi}{4}$$