High School Math : Understanding Maclaurin Series

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : Understanding Maclaurin Series

Give the \displaystyle x^{4} term of the Maclaurin series expansion of the function \displaystyle f(x) = x \tan^{-1} x.

Possible Answers:

\displaystyle - \frac{3}{4} x^{4 }

\displaystyle - \frac{1}{3} x^{4 }

\displaystyle 0

\displaystyle - \frac{1}{4} x^{4 }

\displaystyle - \frac{1}{12} x^{4 }

Correct answer:

\displaystyle - \frac{1}{3} x^{4 }

Explanation:

This can most easily be answered by recalling that the Maclaurin series for \displaystyle \tan^{-1} x is 

\displaystyle \tan^{-1} x = x - \frac{1}{3} x^{3 } + \frac{1}{5} x^{5 }...

Multiply by \displaystyle x to get:

 \displaystyle x\tan^{-1} x = x\cdot x - x\cdot \frac{1}{3} x^{3 } + x\cdot \frac{1}{5} x^{5 }...

\displaystyle x \tan^{-1} x=x^{2 } - \frac{1}{3} x^{4 } + \frac{1}{5} x^{6 }...

The \displaystyle x^{4} term is therefore \displaystyle - \frac{1}{3} x^{4 }.

Example Question #2 : Understanding Maclaurin Series

Give the \displaystyle x^ {3} term of the Maclaurin series of the function \displaystyle f(x) = \sinh 3x .

Possible Answers:

\displaystyle -9x^{3}

\displaystyle 9x^{3}

\displaystyle \frac{9}{2} x^{3}

\displaystyle -\frac{9}{2} x^{3}

\displaystyle 0

Correct answer:

\displaystyle \frac{9}{2} x^{3}

Explanation:

The \displaystyle x^ {3} term of a Maclaurin series expansion has coefficient

\displaystyle \frac{f ' '' (0)}{3!} = \frac{f ' '' (0)}{6}.

We can find \displaystyle f'''(x) by differentiating three times in succession:

\displaystyle f(x) = \sinh 3x

\displaystyle f'(x) = 3 \cosh 3x

\displaystyle f''(x) = 3\cdot 3\sinh 3x = 9\sinh 3x

\displaystyle f'''(x) = 9 \cdot 3\cosh 3x = 27 \cosh 3x

\displaystyle f'''(0) = 27 \cosh \left (3 \cdot 0 \right ) = 27 \cosh 0 = 27\cdot 1 = 27

The term we want is therefore

\displaystyle \frac{f ' '' (0)}{6} \cdot x^{3} = \frac{27}{6} \cdot x^{3} = \frac{9}{2} x^{3}

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