Unit 1: Sequences & Series TestsStudent ID:

Day 16 – The Direct Comparison Test

Opener:Given the function , and knowing that for all values of , then:

Sketch a potential graph of both and / has an asymptote. Sketch a potential graph of both and

Lesson:The Direct ______Test uses one series for which we know the convergence/divergence and compares that series against another one with similar behavior (think back to the opener). To begin, imagine we have two sequences and . There are four scenarios which will help us compare the series below:


Because: /
Because:

Because: /
Because:

The tricky part about the Direct Comparison test is you need to CHOOSE a function (either or ) where you not only know its convergence/divergence, but also know that either it is boundedby the other function or it bounds the other function.

Bounding is an interesting concept in math. While true that an asymptote bounds a function, that is not the only way a function can be bounded. Take, for example, the equation . This function is bounded above by and below by . The technical definition for a bound is:

or is bounded by a number such that for all “relevant” (think infinitely large) values of

Example 1: Use the Direct Comparison Test to identify the convergence of the series


Step 1: Choose a series that behaves in a similar fashion to the one above, but for which you actually know its convergence or divergence.

Step 2: Decide which sequence bounds the other. Do this by listing the first few terms of each sequence. Then, state the bounding of the two functions using inequalities.
Step 3: Use your comparison series from Step 1 to (potentially) conclude convergence or divergence of the original series. State that you used the Direct Comparison Test.

Often times, the function you are asked to find convergence for has “extra” coefficients and terms which don’t really affect the overall convergence. Your job is to figure out how to strip those extra terms away so that you have a simpler function to use as a comparison. Let’s try another:

Example 2:Use the Comparison Test to determine whether the following series will converge.

Example 3: Use the Comparison Test to determine whether the following series will converge.

Of course, we need to be careful. It is not always easy to find a series to compare with the original given. Also, the obvious comparison may not always work! Take a look at a very similar problem to Example 2.

Example 4:Use the Comparison Test to determine whether the following series will converge.

We now have seven types of series and/or series tests. As a challenge, list each of the seven below. Then, give a short definition about how you can identify the sum, or convergence, using each one.

Bookwork:p.650 #s 27 – 33 odds