Taylor's and Maclaurin's Series

Learning Objectives:

  • State the definition of the Taylor/McLaurin series of a function and describe its properties.

  • To understand how Taylor polynomials can be used to approximate functions.

Taylor’s Series is defined as:

\[{f\left( x \right) }={ \sum\limits_{n = 0}^\infty {{f^{\left( n \right)}}\left( a \right)\dfrac{{{{\left( {x - a} \right)}^n}}}{{n!}}} }\] \[= {f\left( a \right) + f'\left( a \right)\left( {x - a} \right) }+{ \dfrac{{f^{\prime\prime}\left( a \right){{\left( {x - a} \right)}^2}}}{{2!}} + \ldots } + {\dfrac{{{f^{\left( n \right)}}\left( a \right){{\left( {x - a} \right)}^n}}}{{n!}} }+...\]

If a=0, the series is called Maclaurin’s Series:

\[{f\left( x \right) }={ \sum\limits_{n = 0}^\infty {{f^{\left( n \right)}}\left( 0 \right)\dfrac{{{x^n}}}{{n!}}} } = {f\left( 0 \right) + f'\left( 0 \right)x }+{ \dfrac{{f^{\prime\prime}\left( 0 \right){x^2}}}{{2!}} + \ldots } +{ \dfrac{{{f^{\left( n \right)}}\left( 0 \right){x^n}}}{{n!}} }+...\]

Some Important MacLaurin’s Series

\[{{e^x} = \sum\limits_{n = 0}^\infty {\dfrac{{{x^n}}}{{n!}}} }={ 1 + x + {\dfrac{{{x^2}}}{{2!}}} }+{ {\dfrac{{{x^3}}}{{3!}}} + ...}\] \[{\cos x = \sum\limits_{n = 0}^\infty {\dfrac{{{{\left( { - 1} \right)}^n}{x^{2n}}}}{{\left( {2n} \right)!}}} }={ 1 - {\dfrac{{{x^2}}}{{2!}}} }+{ {\dfrac{{{x^4}}}{{4!}}} }-{ {\dfrac{{{x^6}}}{{6!}}} + ... }\] \[{\sin x = \sum\limits_{n = 0}^\infty {\dfrac{{{{\left( { - 1} \right)}^n}{x^{2n + 1}}}}{{\left( {2n + 1} \right)!}}} }={ x - {\dfrac{{{x^3}}}{{3!}}} }+{ {\dfrac{{{x^5}}}{{5!}}} }-{ {\dfrac{{{x^7}}}{{7!}}} + ... }\] \[{\cosh x = \sum\limits_{n = 0}^\infty {\dfrac{{{x^{2n}}}}{{\left( {2n} \right)!}}} }={ 1 + {\dfrac{{{x^2}}}{{2!}}} + {\dfrac{{{x^4}}}{{4!}}} }+{ {\dfrac{{{x^6}}}{{6!}}} + ... }\] \[{\sinh x = \sum\limits_{n = 0}^\infty {\dfrac{{{x^{2n + 1}}}}{{\left( {2n + 1} \right)!}}} }={ x + {\dfrac{{{x^3}}}{{3!}}} }+{ {\dfrac{{{x^5}}}{{5!}}} }+{ {\dfrac{{{x^7}}}{{7!}}} + ... }\]

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