Creating power series from geometric series using differentiation and integration

The  exercise appears under the Integral calculus Math Mission. This exercise creates power series from other known power series using derivatives and antiderivatives.

Types of Problems
There are two types of problems in this exercise:


 * 1) Find the power series of the function: This problem provides a rational function. The student is asked to take that function and find it's power series expansion and use it to answer a question.Cpsfgsudai1.png
 * 2) Find the function for the power series: This problem provides the power series of some function. The student is expected to find the function that is being represented and use it to answer a question.Cpsfgsudai2.png

Strategies
Knowledge of the Taylor's theorem and infinite geometric series are encouraged to ensure success on this exercise.
 * 1) The sum of an infinite geometric series is \frac{g_n}{1-r} but can only be used if the magnitude of the ratio is less than one.
 * 2) Each function should be a derivative or antiderivative of some rational function, so that the geometric series exercise can be used as a stepping stone.

Real-life Applications

 * 1) Real-life situations are not "exact" so approximating functions are used as models of real-life behavior in the sciences and economics. These concepts are used to create those approximating functions.