How do you find the Maclaurin series for #1/(1+x^3)#?

1 Answer
Bill K.
Dec 26, 2016

#1-x^3+x^6-x^9+x^12-x^15+cdots# for #|x|<1#.

Explanation:

The quickest way to derive the answer is to view #1/(1+x^3)# as the sum of a geometric series. That is, use the formula #a+ar+ar^2+ar^3+cdots=a/(1-r)# when #|r|<1#.

In the given situation, #a=1# and #r=-x^3#. Therefore,
#1/(1+x^3)=1-x^3+x^6-x^9+x^12-x^15+cdots# for #|-x^3|<1#, which is equivalent to #|x|<1#

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