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[parent] Maclaurin series examples (Example)

Maclaurin series examples

Example 1

One of the simplest Machlaurin series examples is the function

$\displaystyle f\left(x\right) =\frac{1}{1-x} = \left(1-x\right)^{-1}$ (1)

Apply the chain rule to get the derivatives

$\displaystyle f^{\prime}(x) = -\left(1-x\right)^{-2}(-1) = \frac{1}{\left(1-x\right)^2}$

$\displaystyle f^{\prime\prime}(x) = -2\left(1-x\right)^{-3}(-1) = \frac{2}{\left(1-x\right)^3}$

$\displaystyle f^{\prime\prime\prime}(x) = -6\left(1-x\right)^{-4}(-1) = \frac{6}{\left(1-x\right)^4}$

Evaluating the function and its derivatives at zero yields

$\displaystyle f(0) = \frac{1}{1-0} = 1$

$\displaystyle f^{\prime}(0) = \frac{1}{\left(1-0\right)^2} = 1$

$\displaystyle f^{\prime\prime}(0) = \frac{2}{\left(1-0\right)^3} = 2$

$\displaystyle f^{\prime\prime\prime}(0) = \frac{6}{\left(1-0\right)^4} = 6$

Using the formula for the Machlaurin series

$\displaystyle f\left(x\right) = \sum _{m=0}^{\infty }\frac{f^{\left(m\right)}\l... ...\left(0\right)}{2!}x^2+\frac{f^{\prime\prime\prime}\left(0\right)}{3!}x^3+\dots$ (2)
$\displaystyle f\left(x\right) = \sum _{m=0}^{\infty }\frac{f^{\left(m\right)}\left(0\right)}{m!}x^m=1+\frac{1}{1!}x+\frac{2}{2!}x^2+ \frac{6}{3!}x^3\dots$ (3)

Finally, evaluating the factorials and seeing they divide out the numerator we get the series

$\displaystyle f\left(x\right) = 1+x+x^2+x^3\dots$ (4)

Let's evaluate example 1 function at $x=0.5$ and see how the order of the series (powers of x, so order 3 would go through $x^3$

Figure: Example 1 at $x=0.5$
\resizebox{\textwidth}{!}{\includegraphics{example1.png}}

Example 2

Another popular example for the Machlaurin series is the function

$\displaystyle f\left(x\right) = sin(x)$ (5)

Get the derivatives, do you see a pattern?

$\displaystyle f^{\prime}(x) = cos(x)$

$\displaystyle f^{\prime\prime}(x) = -sin(x)$

$\displaystyle f^{\prime\prime\prime}(x) = -cos(x)$

$\displaystyle f^{\prime\prime\prime\prime}(x) = sin(x)$

Evaluating the function and its derivatives at zero yields

$\displaystyle f^{\prime}(0) = cos(0) = 1$

$\displaystyle f^{\prime\prime}(0) = -sin(0) = 0$

$\displaystyle f^{\prime\prime\prime}(0) = -cos(x) = -1$

$\displaystyle f^{\prime\prime\prime\prime}(0) = sin(x) = 0$

Using the formula for the Machlaurin series

$\displaystyle f\left(x\right) = \sum _{m=0}^{\infty }\frac{f^{\left(m\right)}\l... ...\left(0\right)}{2!}x^2+\frac{f^{\prime\prime\prime}\left(0\right)}{3!}x^3+\dots$ (6)
$\displaystyle f\left(x\right) = \sum _{m=0}^{\infty }\frac{f^{\left(m\right)}\l... ...^m=0+\frac{1}{1!}x+0- \frac{1}{3!}x^3+0+\frac{1}{5!}x^5+0- \frac{1}{7!}x^7\dots$ (7)

Finally, evaluating the factorials we get the series

$\displaystyle f\left(x\right) = x- \frac{x^3}{3}+\frac{x^5}{120}- \frac{x^7}{5040}\dots$ (8)

Bibliography

[1] Kreyszig, E., “Advanced Engineering Mathematics.” Fifth Edition. John Wiley and Sons, Inc. 1983.



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