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[parent] Maclaurin series examples (Example)

1 Maclaurin series examples

1.1 Example 1

One of the simplest Machlaurin series examples is the function

1 f (x ) = ------= (1 − x)−1 1 − x
(1)

Apply the chain rule to get the derivatives

′ −2 1 f (x) = − (1 − x) (− 1) = -------2 (1 − x)

−3 2 f′′(x ) = − 2 (1 − x) (− 1) = -------3 (1 − x)

6 f′′′(x) = − 6 (1 − x)−4(− 1) =-------4 (1 − x)

Evaluating the function and its derivatives at zero yields

1 f (0 ) = ------= 1 1 − 0

′ ---1---- f(0) = (1 − 0)2 = 1

′′ 2 f (0) = -------3 = 2 (1 − 0)

f ′′′(0) = ---6---- = 6 (1 − 0)4

Using the formula for the Machlaurin series

∑∞ f(m)(0) f′(0) f ′′(0) f ′′′(0 ) f (x) = -------xm = f (0) + -----x + ------x2 + -------x3 + ... m=0 m! 1! 2! 3!
(2)

∞ ∑ f(m)(0)-m 1- -2 2 -6 3 f (x ) = m! x = 1 + 1!x + 2!x + 3!x ... m=0
(3)

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

f (x ) = 1 + x + x2 + x3 ...
(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 x3

PIC

Figure 1. Example 1 at x = 0.5.

1.2 Example 2

Another popular example for the Machlaurin series is the function

f (x ) = sin (x )
(5)

Get the derivatives, do you see a pattern?

f′(x ) = cos (x )

f ′′(x) = − sin (x)

f ′′′(x) = − cos(x)

f′′′′(x) = sin(x)

Evaluating the function and its derivatives at zero yields

′ f (0 ) = cos (0 ) = 1

f′′(0) = − sin (0) = 0

′′′ f (0) = − cos(x) = − 1

′′′′ f (0) = sin(x) = 0

Using the formula for the Machlaurin series

∞ ∑ f(m)(0)- m f′(0)- f-′′(0) 2 f-′′′(0-) 3 f (x) = m! x = f (0) + 1! x + 2! x + 3! x + ... m=0
(6)

∑∞ (m ) f (x) = f----(0-)xm = 0 + 1-x + 0 − 1-x3 + 0 + 1-x5 + 0 − 1-x7... m! 1! 3! 5! 7! m=0
(7)

Finally, evaluating the factorials we get the series

x3- -x5- -x7-- f (x ) = x − 3 + 120 − 5040 ...
(8)

References

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


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See Also: Applications of the Maclaurin Series in Physical Modeling


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Cross-references: powers, formula, function

This is version 5 of Maclaurin series examples, born on 2025-07-14, modified 2026-08-16.
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Physics Classification: 02.30.-f (Function theory, analysis)
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