Tuesday, July 26, 2016

calculus - Maclaurin polynomial of tan(x)



The method used to find the Maclaurin polynomial of sin(x), cos(x), and ex requires finding several derivatives of the function. However, you can only take a couple derivatives of tan(x) before it becomes unbearable to calculate.




Is there a relatively easy way to find the Maclaurin polynomial of tan(x)?



I considered using tan(x)=sin(x)/cos(x) somehow, but I couldn't figure out how.


Answer



Long division of series.



x+x33+2x515+1x22+x424+)¯xx36+x5120+xx32+x524+x33x530+x33x56+2x515+2x515+


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