Final answer:
The question concerns finding the power series of the function 1/(1+x)(1-2x), which can be represented by a Taylor or Maclaurin series, and potentially by a geometric series after suitable manipulation.
Step-by-step explanation:
The student is asking about finding the power series of the function 1/(1+x)(1-2x). Three types of series mentioned in the question—a Taylor series, a geometric series, and a Maclaurin series—are all based on the concept of representing functions as an infinite sum of terms calculated from the values of the function's derivatives at a single point. For a Maclaurin series, this point is typically at x = 0.
The geometric series can also be used to find the power series of the given function. For a function to be expressed as a geometric series, it must be in the form 1/(1-r), where r is the common ratio. In this case, the function would need to be manipulated so that each component matches the geometric series form. The binomial theorem can provide assistance in expressing the modified structure of (1+x) and (1-2x) to power series.
It is important to note that a rational series is not a commonly used term in the context of series expansions, so it seems to be a misnomer or extraneous option in the question.