Maclaurin Series

The Maclaurin series of a function f is the series

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

where f(n)(0) indicates the nth derivative of f evaluated at 0.

To expand a function f (x) in a Maclaurin series (power series about x = 0), choose Power Series, specify desired Number of Terms, and then specify Expand in Powers of x. With f (x) = ${\frac{{\sin x}}{{x}}}$ and 10 terms, the result is as follows.

$\blacktriangleright$ Power Series

${\dfrac{{\sin x}}{{x}}}$ = 1 - ${\frac{{1}}{{6}}}$x2 + ${\frac{{1}}{{120}}}$x4 - ${\frac{{1}}{{5040}}}$x6 + ${\frac{{1}}{{362880}}}$x8 + O$\left(\vphantom{ x^{9}}\right.$x9$\left.\vphantom{ x^{9}}\right)$

The O$\left(\vphantom{ x^{9}}\right.$x9$\left.\vphantom{ x^{9}}\right)$ term indicates that all the remaining terms in the series contain at least x9 as a factor. (In fact, the truncation error is of order x10 in this case.) The odd powers of x have coefficients of 0.

Plot 2D provides an excellent visual comparison between a function and an approximating polynomial.

$\blacktriangleright$ Plot 2D + Rectangular

${\dfrac{{\sin x}}{{x}}}$, 1 - ${\frac{{1}}{{6}}}$x2 + ${\frac{{1}}{{120}}}$x4

dtbpF3in2.0003in0pt

To determine which graph corresponds to which equation, evaluate one of the expressions where the graphs show some separation. For example, ${\frac{{\sin 4}}{{4}}}$ = - .1892006238, and hence the graph of ${\frac{{\sin x}}{{x}}}$ is the one that is negative at x = 4.

The following are additional examples of Maclaurin series expansions.

$\blacktriangleright$ Power Series

ln(1 - x) = - x - ${\frac{{1}}{{2}}}$x2 - ${\frac{{1}}{{3}}}$x3 - ${\frac{{1}}{{4}}}$x4 - ${\frac{{1}}{{5}}}$x5 + O$\left(\vphantom{ x^{6}}\right.$x6$\left.\vphantom{ x^{6}}\right)$

tan-1x = x - ${\frac{{1}}{{3}}}$x3 + ${\frac{{1}}{{5}}}$x5 - ${\frac{{1}}{{7}%
}}$x7 + ${\frac{{1}}{{9}}}$x9 + O$\left(\vphantom{ x^{10}}\right.$x10$\left.\vphantom{ x^{10}}\right)$

sin2x = x2 - ${\frac{{1}}{{3}}}$x4 + ${\frac{{2}}{{45}}}$x6 - ${\frac{{1}}{{315}}}$x8 + O$\left(\vphantom{ x^{10}}\right.$x10$\left.\vphantom{ x^{10}}\right)$

ex = 1 + x + ${\frac{{1}}{{2}}}$x2 + ${\frac{{1}}{{6}}}$x3 + ${\frac{{1}}{{24}}}$x4 + ${\frac{{1}}{{120}}}$x5 + ${\frac{{1}}{{720}}}$x6 + O$\left(\vphantom{ x^{7}}\right.$x7$\left.\vphantom{ x^{7}}\right)$

sin x = x - ${\frac{{1}}{{6}}}$x3 + ${\frac{{1}}{{120}}}$x5 - ${\frac{{1}}{{5040}%
}}$x7 + ${\frac{{1}}{{362880}}}$x9 + O$\left(\vphantom{ x^{10}}\right.$x10$\left.\vphantom{ x^{10}}\right)$

exsin x = x + x2 + ${\frac{{1}}{{3}}}$x3 - ${\frac{{1}}{{30}}}$x5 - ${\frac{{1}}{{%
90}}}$x6 + O$\left(\vphantom{ x^{7}}\right.$x7$\left.\vphantom{ x^{7}}\right)$

Remember that output can be copied and pasted (with ordinary word-processing tools) to create input for further calculations. In particular, select and delete the + O$\left(\vphantom{ x^{n}}\right.$xn$\left.\vphantom{ x^{n}}\right)$ expression to convert the series into a polynomial. It is reassuring to note that, if the first few terms of the Maclaurin series for ex are multiplied by the first few terms of the Maclaurin series for sin x, then the result is the same as the first few terms of the Maclaurin series for exsin x.

$\blacktriangleright$ Expand, Polynomials + Sort

$\left(\vphantom{ 1+x+\frac{1}{2}x^{2}+\frac{1}{6}x^{3}+\frac{1}{24}x^{4}+\frac{1}{120}%
x^{5}}\right.$1 + x + ${\frac{{1}}{{2}}}$x2 + ${\frac{{1}}{{6}}}$x3 + ${\frac{{1}}{{24}}}$x4 + ${\frac{{1}}{{120}%
}}$x5$\left.\vphantom{ 1+x+\frac{1}{2}x^{2}+\frac{1}{6}x^{3}+\frac{1}{24}x^{4}+\frac{1}{120}%
x^{5}}\right)$$\left(\vphantom{ \allowbreak x-\frac{1}{6}x^{3}+\frac{1}{120}x^{5}}\right.$x - ${\frac{{1}}{{6}}}$x3 + ${\frac{{1}}{{120}}}$x5$\left.\vphantom{ \allowbreak x-\frac{1}{6}x^{3}+\frac{1}{120}x^{5}}\right)$

= x + ${\frac{{1}}{{3}}}$x3 - ${\frac{{1}}{{30}}}$x5 + x2 - ${\frac{{1}}{{90}}}$x6 - ${\frac{{1}}{{360}}}$x7 + ${\frac{{1}}{{2880}}}$x9 + ${\frac{{1}}{{14400}}}$x10

= ${\frac{{1}}{{14400}}}$x10 + ${\frac{{1}}{{2880}}}$x9 - ${\frac{{1}}{{360}}}$x7 - ${\frac{{1}}{{90}}}$x6 - ${\frac{{1}}{{30}}}$x5 + ${\frac{{1}}{{3}}}$x3 + x2 + x