The [i]Maclaurin series[/i] is a template that allows you to express many other functions as power series. It is the source of formulas for expressing both sin [i]x [/i]and cos [i]x[/i] as infinite series.Without further ado, here it is:[br][br][img width=171,height=56]https://www.dummies.com/wp-content/uploads/315812.image0.png[/img][br][br]The notation [i]f[/i]([i]n[/i]) means “the [i]n[/i]th derivative of [i]f[/i][i].[/i]” This becomes clearer in the expanded version of the Maclaurin series:[br][br][img width=401,height=44]https://www.dummies.com/wp-content/uploads/315813.image1.png[/img][br][br]The Maclaurin series allows you to express functions as power series by following these steps:[list=1][*]Find the first few derivatives of the function until you recognize a pattern.[/*][*]Substitute 0 for [i]x [/i]into each of these derivatives.[/*][*]Plug these values, term by term, into the formula for the Maclaurin series.[/*][*]If possible, express the series in sigma notation.[/*][/list]

For instance, suppose you were interested in finding the power series representation of[br][br][br][img width=109,height=40]http://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/SandS/PowerSeries/Functions/example.gif[/img][br][br]We can find the power representation of this function like so:[br][br][img width=204,height=40]http://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/SandS/PowerSeries/Functions/example_deriv.gif[/img],[br][br]so[br][br]r[img width=123,height=34]http://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/SandS/PowerSeries/Functions/example_derivative2.gif[/img].[img width=308,height=39]http://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/SandS/PowerSeries/Functions/example2.gif[/img],[br][br][br]where [i]y[/i]=[i]x[/i]/3, so [img width=268,height=43]http://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/SandS/PowerSeries/Functions/example_sum.gif[/img].[br][br][br][br]Thus,[br][br][img width=378,height=43]http://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/SandS/PowerSeries/Functions/example_done1.gif[/img][img width=312,height=46]http://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/SandS/PowerSeries/Functions/example_done2.gif[/img][br][br][br][br]