Taylor Series

The Maclaurin series is a special case of the more general Taylor series. The Taylor series of f expanded about x = a is given by

$\displaystyle \sum_{{n=0}}^{{\infty }%
}$$\displaystyle {\frac{{f^{(n)}(a)}}{{n!}}}$(x - a)n

and hence is expanded in powers of x - a.

For Taylor series, enter the number of terms and the point of expansion in the Series dialog box. To find the Taylor series of ln x expanded about x = 1, choose Powers Series. In the dialog box, select the desired number of terms and expand about the point x - 1.

$\blacktriangleright$ Power Series

ln x = $\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)$ - ${\frac{{1}}{{2}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{2}}_{}$ + ${\frac{{1}}{{3}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{3}}_{}$ - ${\frac{{1}}{{4}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{4}}_{}$ + O$\left(\vphantom{
\left( x-1\right) ^{5}}\right.$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{5}}_{}$$\left.\vphantom{
\left( x-1\right) ^{5}}\right)$

A comparison between ln x and the polynomial $\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)$ - ${\frac{{1}}{{2}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{2}}_{}$ + ${\frac{{1}}{{3}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{3}}_{}$ - ${\frac{{1}}{{4}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{4}}_{}$ is illustrated graphically in the following figure.

$\blacktriangleright$ Plot 2D + Rectangular

ln x,$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)$ - ${\frac{{1}}{{2}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{2}}_{}$ + ${\frac{{1}}{{3}%
}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{3}}_{}$ - ${\frac{{1}}{{4}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{4}}_{}$

dtbpF3in2.0003in0pt

You can produce the following power series expansions with Scientific Notebook.

$\blacktriangleright$ Power Series

${\frac{{1}}{{x}}}$ = ${\frac{{1}}{{2}}}$ - ${\frac{{1}}{{4}}}$$\left(\vphantom{ x-2}\right.$x - 2$\left.\vphantom{ x-2}\right)$ + ${\frac{{1}}{{8}}}$$\left(\vphantom{ x-2}\right.$x - 2$\left.\vphantom{ x-2}\right)^{{2}}_{}$ - ${\frac{{1}}{{16}}}$$\left(\vphantom{ x-2}\right.$x - 2$\left.\vphantom{ x-2}\right)^{{3}}_{}$ + ${\frac{{1}}{{32}%
}}$$\left(\vphantom{ x-2}\right.$x - 2$\left.\vphantom{ x-2}\right)^{{4}}_{}$ + O$\left(\vphantom{ \left( x-2\right) ^{5}}\right.$$\left(\vphantom{ x-2}\right.$x - 2$\left.\vphantom{ x-2}\right)^{{5}}_{}$$\left.\vphantom{ \left( x-2\right) ^{5}}\right)$

sin x = - $\left(\vphantom{ x-\pi }\right.$x - π$\left.\vphantom{ x-\pi }\right)$ + ${\frac{{1}}{{6}}}$$\left(\vphantom{ x-\pi }\right.$x - π$\left.\vphantom{ x-\pi }\right)^{{3}}_{}$ + O$\left(\vphantom{ \left( x-\pi \right) ^{5}}\right.$$\left(\vphantom{ x-\pi }\right.$x - π$\left.\vphantom{ x-\pi }\right)^{{5}}_{}$$\left.\vphantom{ \left( x-\pi \right) ^{5}}\right)$

$\sqrt{{x}}$ = 1 + ${\frac{{1}}{{2}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)$ - ${\frac{{1}}{{8}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{2}}_{}$ + ${\frac{{1}}{{16}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{3}}_{}$ - ${\frac{{5}}{{128}}}$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{4}}_{}$ + O$\left(\vphantom{
\left( x-1\right) ^{5}}\right.$$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)^{{5}}_{}$$\left.\vphantom{
\left( x-1\right) ^{5}}\right)$

csc x = 1 + ${\frac{{1}}{{2}}}$$\left(\vphantom{ x-\frac{1}{2}\pi }\right.$x - ${\frac{{1}}{{2}}}$π$\left.\vphantom{ x-\frac{1}{2}\pi }\right)^{{2}}_{}$ + ${\frac{{5}}{{24}}}$$\left(\vphantom{ x-\frac{1}{2}\pi }\right.$x - ${\frac{{1}}{{2}}}$π$\left.\vphantom{ x-\frac{1}{2}\pi }\right)^{{4}}_{}$ + O$\left(\vphantom{ \left( x-\frac{1}{2}\pi
\right) ^{5}}\right.$$\left(\vphantom{ x-\frac{1}{2}\pi }\right.$x - ${\frac{{1}}{{2}}}$π$\left.\vphantom{ x-\frac{1}{2}\pi
}\right)^{{5}}_{}$$\left.\vphantom{ \left( x-\frac{1}{2}\pi
\right) ^{5}}\right)$

2 sin2x = 2$\left(\vphantom{ x-\pi }\right.$x - π$\left.\vphantom{ x-\pi }\right)^{{2}}_{}$ - ${\frac{{2}}{{3}}}$$\left(\vphantom{ x-\pi }\right.$x - π$\left.\vphantom{ x-\pi
}\right)^{{4}}_{}$ + O$\left(\vphantom{ \left( x-\pi \right) ^{6}}\right.$$\left(\vphantom{ x-\pi }\right.$x - π$\left.\vphantom{ x-\pi }\right)^{{6}}_{}$$\left.\vphantom{ \left( x-\pi \right) ^{6}}\right)$

1 - cos 2x = 2$\left(\vphantom{ x-\pi }\right.$x - π$\left.\vphantom{ x-\pi }\right)^{{2}}_{}$ - ${\frac{{2}}{{3}}}$$\left(\vphantom{ x-\pi }\right.$x - π$\left.\vphantom{ x-\pi
}\right)^{{4}}_{}$ + O$\left(\vphantom{ \left( x-\pi \right) ^{5}}\right.$$\left(\vphantom{ x-\pi }\right.$x - π$\left.\vphantom{ x-\pi }\right)^{{5}}_{}$$\left.\vphantom{ \left( x-\pi \right) ^{5}}\right)$