Infinite geometric series

The Infinite geometric series exercise appears under the Precalculus Math Mission, Mathematics III Math Mission and Integral calculus Math Mission. This exercise uses a plethora of problems to explore the sum of infinite geometric sequences.

Types of Problems

There are six types of problems in this exercise:

1. Find the sum: This problem provides an infinite sum. The user is expected to find the correct answer and write it in the space provided.

   

   Find the sum

2. Given the sum, find other information: This problem provides the sum of an infinite geometric sequence and some other information. The user is tasked with finding some other piece of missing information based on the givens.

   

   Given the sum, find other information

3. Solve the word problem: This problem provides a real-life application that involves an infinite geometric sum. The user is asked to solve the problem and provide the correct answer.

   

   Solve the word problem

4. Evaluate sigma notation: This problem also has the user find an infinite geometric sum, this time, written in the sigma notation.

5. 

   Evaluate sigma notation

   Determine convergence: This problem provides several geometric series. The user is asked to select all of the series that converge from the multiple select list.

   

   Determine convergence

6. Find the function representation: This problem provides an infinite geometric series that can be written as a continuous function in x. The user is asked to determine which function is being represented by the sum.

   

   Find the function representation

Strategies

Knowledge of geometric sequences and beginning ideas of convergence are encouraged to ensure success on this exercise.

1. The infinite geometric series formula is ${S_\infty=\frac{g_1}{1-r}}$.
2. The infinite geometric series is the limit as n goes to infinity of the finite geometric series formula. It only converges if the magnitude of the ratio is less than one.
3. If the common ratio is -1, the series will tend to oscillate, and it diverges to infinity if the magnitude of the ratio is greater than one.

Real-life Applications

1. Summation (or sigma) notation is a notation used for representing long sums.
2. The concepts in this exercise show up in second semester calculus as related to Taylor polynomials.
3. There are some real-life applications among the problems on this exercise.
4. The distance travelled by a bouncing ball is a classic application of this concept.