Creating power series from geometric series using algebra

The Creating power series from geometric series using algebra exercise appears under the Integral calculus Math Mission. This exercise creates power series from geometric series.

Types of Problems

There are three types of problems in this exercise:

1. Find the interval of convergence for the power series: This problem provides a power series. The user is asked to find the domain on which the series is known to converge, also called the interval of convergence.

   

   Find the interval of convergence for the power series

2. Find the power series of the function: This problem provides a rational function. The user is asked to take that function and find it's power series expansion and use it to answer a question.

   

   Find the power series of the function

3. Find the function for the power series: This problem provides the power series of some function. The user is expected to find the function that is being represented and use it to answer a question.

   

   Find the function for the power series

Strategies

Knowledge of Taylor's theorem and infinite geometric series are encouraged to ensure success on this exercise.

1. The ratio test can be used to find the interval of convergence. The answer for ${\frac{x-a}{b}}$ is usually an interval centered at ${a}$ with radius ${b}$. The endpoints can usually be checked with an alternating series test.
2. When finding the value of an infinite series, an appropriate function can change the question into evaluating ${a}$ function.
3. 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.

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.
2. Interval of convergence is used with Taylor's series.