Alternating series

The  exercise appears under the Integral calculus Math Mission. This exercise practices one of the series convergence/divergence tests and plays with the idea of conditional convergence of series.

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


 * 1) Determine if sequence converges conditionally or absolutely: This problem provides a groups of sequences that are being summed. The student is asked to determine the convergence of each series, then use the result to tell if a particular series converges conditionally or absolutely.Altsr1.png
 * 2) Determine the values for the variable to allow convergence: This problem has a series with a constant variable. The student is asked to determine bounds on the constant to ensure convergence of the series.Altsr2.png

Strategies
Knowledge of all limit tests, but specifically the alternating series test, as well as experience with the limit of a sequence are encouraged to ensure success on this exercise.
 * 1) The alternating series tests states that if a sequence converges to zero, and it alternates positive and negative, then it converges. However, the convergence can be conditional.
 * 2) If a series and it's absolute value both converge, then the series converges absolutely.
 * 3) If an alternating series converges but it's absolute value does not, then it converges conditionally.

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) The harmonic series, with and alternator, converges conditionally.
 * 3) When a series converges conditionally, although there is a limit, the terms of sequence can be rearranged to create any limit that is desired. It is very useful for counterexamples in higher mathematics.