Direct and limit comparison tests

The  exercise appears under the Integral calculus Math Mission. This exercise practices one of the series convergence/divergence tests.

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


 * 1) Apply the "direct" comparison test: This problem provides a series that is similar to a series that is known to converge or diverge. The student is expected to directly compare the given series to the known series and determine convergence.Dalct1.png
 * 2) Apply the "limit" comparison test: This problem provides a series that can be compared with another series that is known to converge or diverge. The student is expected to compare the given series, in limit, to the known series and determine convergence.Dalct2.png
 * 3) Determine the appropriate test and conclusion: This problem provides a series and no indication of the test that is used. The student is asked to determine which test would be appropriate and what the result of that test would be.Dalct3.png

Strategies
Knowledge of all limit tests, but specifically the comparison tests, as well as experience with the limit of a sequence are encouraged to ensure success on this exercise.
 * 1) The "direct" comparison test often compares a series to either 1/n or 1/n^2. If it is larger than a divergent series, it diverges. If it is less than a convergent series, it converges.
 * 2) The "limit" comparison test finds the limit of the ratio of two sequences. If the result is finite-positive, both series diverge or both converge. If it goes to 0 and the bottom converges, then so does the top. If it goes to infinity and the bottom diverges, so does the top.
 * 3) The series in the question can be used to motivate a comparison series.

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) Knowing how to compare unknown things with known things is a common exercise in mathematics, with applications over many fields.