6.4 Working with taylor series  (Page 7/11)

Use ${\left(1+x\right)}^{1\text{/}3}=1+\frac{1}{3}x-\frac{1}{9}{x}^2+\frac{5}{81}{x}^3-\frac{10}{243}{x}^4+\text{⋯}$ with $x=1$ to approximate $2^{1\text{/}3}.$

Use the approximation ${\left(1-x\right)}^{2\text{/}3}=1-\frac{2x}{3}-\frac{x^2}{9}-\frac{4x^3}{81}-\frac{7x^4}{243}-\frac{14x^5}{729}+\text{⋯}$ for $\left|x\right|<1$ to approximate $2^{1\text{/}3}={2.2}^{\mathrm{-2}\text{/}3}.$

Find the $25\text{th}$ derivative of $f\left(x\right)={\left(1+x^2\right)}^{13}$ at $x=0.$

Find the $99$ th derivative of $f\left(x\right)={\left(1+x^4\right)}^{25}.$

$f\left(x\right)=x{e}^{2x}$

$f\left(x\right)=2^x$

$f\left(x\right)=\frac{\text{sin}x}{x}$

$f\left(x\right)=\frac{\text{sin}\left(\sqrt{x}\right)}{\sqrt{x}},\left(x>0\right),$

$f\left(x\right)=\text{sin}\left(x^2\right)$

$f\left(x\right)=e^{x^3}$

$f\left(x\right)={\text{cos}}^{2}x$ using the identity ${\text{cos}}^{2}x=\frac{1}{2}+\frac{1}{2}\text{cos}\left(2x\right)$

$f\left(x\right)={\text{sin}}^{2}x$ using the identity ${\text{sin}}^{2}x=\frac{1}{2}-\frac{1}{2}\text{cos}\left(2x\right)$

$F\left(x\right)={\displaystyle {\int }_0^x{e}^{\text{−}{t}^2}}dt;f\left(t\right)=e^{\text{−}{t}^2}={\displaystyle \sum _{n=0}^{\infty }{\left(\mathrm{-1}\right)}^n}\frac{t^{2n}}{n\text{!}}$

$F\left(x\right)={\text{tan}}^{\mathrm{-1}}x;f\left(t\right)=\frac{1}{1+t^2}={\displaystyle \sum _{n=0}^{\infty }{\left(\mathrm{-1}\right)}^n{t}^{2n}}$

$F\left(x\right)={\text{tanh}}^{\mathrm{-1}}x;f\left(t\right)=\frac{1}{1-t^2}={\displaystyle \sum _{n=0}^{\infty }{t}^{2n}}$

$F\left(x\right)={\text{sin}}^{\mathrm{-1}}x;f\left(t\right)=\frac{1}{\sqrt{1-t^2}}={\displaystyle \sum _{k=0}^{\infty }\left(\begin{array}{c}\frac{1}{2}\hfill \\ k\hfill \end{array}\right)\frac{t^{2k}}{k\text{!}}}$

$F\left(x\right)={\displaystyle {\int }_0^x\frac{\text{sin}t}{t}dt};f\left(t\right)=\frac{\text{sin}t}{t}={\displaystyle \sum _{n=0}^{\infty }{\left(\mathrm{-1}\right)}^n\frac{t^{2n}}{\left(2n+1\right)\text{!}}}$

$F\left(x\right)={\displaystyle {\int }_0^x\text{cos}\left(\sqrt{t}\right)dt};f\left(t\right)={\displaystyle \sum _{n=0}^{\infty }{\left(\mathrm{-1}\right)}^n\frac{x^n}{\left(2n\right)\text{!}}}$

$F\left(x\right)={\displaystyle {\int }_0^x\frac{1-\text{cos}t}{t^2}dt};f\left(t\right)=\frac{1-\text{cos}t}{t^2}={\displaystyle \sum _{n=0}^{\infty }{\left(\mathrm{-1}\right)}^n\frac{t^{2n}}{\left(2n+2\right)\text{!}}}$

$F\left(x\right)={\displaystyle {\int }_0^x\frac{\text{ln}\left(1+t\right)}{t}dt};f\left(t\right)={\displaystyle \sum _{n=0}^{\infty }{\left(\mathrm{-1}\right)}^n\frac{t^n}{n+1}}$

$f\left(x\right)=\text{sin}\left(x+\frac{\pi }{4}\right)=\text{sin}x\text{cos}\left(\frac{\pi }{4}\right)+\text{cos}x\text{sin}\left(\frac{\pi }{4}\right)$

$f\left(x\right)=\text{tan}x$

$f\left(x\right)=\text{ln}\left(\text{cos}x\right)$

$f\left(x\right)=e^x\text{cos}x$

$f\left(x\right)=e^{\text{sin}x}$

$f\left(x\right)={\text{sec}}^{2}x$

$f\left(x\right)=\text{tanh}x$

$f\left(x\right)=\frac{\text{tan}\sqrt{x}}{\sqrt{x}}$ (see expansion for $\text{tan}x)$

$\text{ln}\left(1+x\right)$

$\frac{1}{1+x^2}$

${\text{tan}}^{\mathrm{-1}}x$

$\text{ln}\left(1+x^2\right)$

Find the Maclaurin series of $\text{sinh}x=\frac{e^x-e^{\text{−}x}}{2}.$

Find the Maclaurin series of $\text{cosh}x=\frac{e^x+e^{\text{−}x}}{2}.$

Differentiate term by term the Maclaurin series of $\text{sinh}x$ and compare the result with the Maclaurin series of $\text{cosh}x.$

[T] Let $S_n\left(x\right)={\displaystyle \sum _{k=0}^n{\left(\mathrm{-1}\right)}^k\frac{x^{2k+1}}{\left(2k+1\right)\text{!}}}$ and $C_n\left(x\right)={\displaystyle \sum _{n=0}^n{\left(\mathrm{-1}\right)}^k\frac{x^{2k}}{\left(2k\right)\text{!}}}$ denote the respective Maclaurin polynomials of degree $2n+1$ of $\text{sin}x$ and degree $2n$ of $\text{cos}x.$ Plot the errors $\frac{S_n\left(x\right)}{C_n\left(x\right)}-\text{tan}x$ for $n=1,\mathrm{..},5$ and compare them to $x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}-\text{tan}x$ on $\left(-\frac{\pi }{4},\frac{\pi }{4}\right).$

Use the identity $2\text{sin}x\text{cos}x=\text{sin}\left(2x\right)$ to find the power series expansion of ${\text{sin}}^{2}x$ at $x=0.$ ( Hint: Integrate the Maclaurin series of $\text{sin}\left(2x\right)$ term by term.)

If $y={\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n},$ find the power series expansions of $x{y}^{\prime }$ and $x^{2}y\text{″}.$

[T] Suppose that $y={\displaystyle \sum _{k=0}^{\infty }{a}_k{x}^k}$ satisfies $y^{\prime }=\mathrm{-2}xy$ and $y\left(0\right)=0.$ Show that $a_{2k+1}=0$ for all $k$ and that $a_{2k+2}=\frac{\text{−}{a}_{2k}}{k+1}.$ Plot the partial sum $S_{20}$ of $y$ on the interval $[\mathrm{-4},4].$