Alternating series remainder

The  exercise appears under the Algebra I Math Mission. This exercise uses the error bound on the alternating series to answer some questions about power series.

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


 * 1) Determine over versus under  and estimate: This problem has an alternating series and a collection of questions. The student is sometimes asked to find the estimate of the error, determine if the approximation is under or over, or some combination of these questions.Asr1.png
 * 2) Find the number of terms to use: This problem has an alternating series that is supposed to be used to estimate some value within a certain degree of accuracy. The student is asked to find the appropriate number of terms of the sequence to use to get the desired accuracy.Asr2.png

Strategies
Knowledge of series notation, alternating series applications, and approximation methods are encouraged to ensure success on this exercise.
 * 1) The bound on the error of a finite series is approximated by the next term in the series.
 * 2) An approximation plus the error is the actual, so a positive error means the approximation was under and a negative error means the approximation was over.
 * 3) To find the needed term to bound the error, set the k+1-th term less than half the desired degree of accuracy.

Real-life Applications

 * 1) Bounds on the approximation of series are useful in real-life where mathematically exact answers are sometimes not possible.
 * 2) Some of the problems in this exercise are applications to power series.