A variety of techniques for showing that sequences have limits can be found, however they are all based in the geometric idea above. If the terms of the sequence eventually all squish together closer than any imaginable distance, then there is convergence. Otherwise there is not.
Then the limit of this sequence is called the sum:
One absolutely indispensible piece of knowledge with regard to series is the ability to quickly recognize and find the sum of a geometric series. In order to give students this ability, the following may be fruitful:
Have students put away all of their materials and listen closely so that they understand every step of what follows. Tell them they will have to do this on their own in a minute, without anything on the board, so they should ask questions if they have any. Then show them the general geometric series:
Then we can subtract these:
and we see that every term will cancel except for the very first r:
to give that:
After letting them think a little, tell them the ANSWER:
Next, the board should be thoroughly erased and students should be asked to reproduce the entire derivation for the sum of a geometric series as well as to solve some similar problem. They can do this individually or in small groups, but the problem should be a little disguised or more complicated such as proving the following result:
Series Without Negative Terms
The harmonic series is extremely important and indeed many mathematicians have dedicated their entire life’s work to understanding it. So it is worthwhile to show students why it clearly diverges. Consider the sum:
And so we see that:
So how many terms can we remove from the harmonic series and still have it diverge? For example, we can remove every term whose denominator is not prime, leaving behind the famous series:
Teaching this material gives some nice opportunities to have students practice with inequalities and integration. By giving series of complicated rational terms the students can try finding an appropriate comparison or integral. In general this will be a daunting and cumbersome activity for students to do on their own, so it is recommended that the class be divided into small groups. Perhaps the following activity would be exciting:
Series With Odd or Even Negative Terms
Series that contain both positive and negative terms should be thought of more likely to converge in a sense. This is because very qualitatively the negative terms will counterbalance the positive ones making the sum more reasonable. For this reason, a series that alternates term-by-term between positive and negative has a very simple test for convergence. If the terms trail off to zero then the series converges. This is clearly not good enough for a strictly positive series, as the harmonic series shows.
This follows from the fact that the signs alternate so that an arrangement of the terms shows that the tail of the series always has the same sign as its leading term. But then since the tail can be written as a sum of terms that all have the same sign, it must be smaller than the leading term if the sign is to come out unchanged.
To teach this material it is good to get students in the habit of writing the first few terms of a series in order to understand its behavior. To this end, a good start to any class on alternating series is to write some series on the board and select students to come up and write out the first few terms of each. This will get everyone on the same page about how these series alternate and what we are really talking about.
absolutely convergent, conditionally convergent, and divergent?
Ratio Test, Root Test, and Summary of Tests
This lesson summarizes a number of different tests for convergence of series. Students will be know that if they understood and can recall how to find the sum of a geometric series then the proofs of these tests will follow. For example, consider the ratio test: Let
The root test has a similar proof except it is in a sense simpler or more direct. If we have that:
This implies that:
This could be taught in a nice way too by having students play a game in teams against one-another:
The idea with this activity is to generate an environment where students are actively working together to seek tests that will demonstrate whether a given series is convergent or divergent. If their tests work then they will know which way to turn, and if not then they will take a wrong turn.
CONTENT The simplest way to present a power series is as an infinitely long polynomial. Just say: “A power series is just an infinitely long polynomial”
The content is more likely to be readily accepted if motivated properly. Therefore it is worthwhile to let students know ahead of time that almost any function there is that is nice enough (differentiable) can be written as an infinite polynomial like this. Since polynomials are so easy to differentiate and integrate, this has enormous utility throughout math, science, business, and engineering.
It’s a good idea to start with some simple examples of power series that are convergent to familiar functions:
Here it is worthwhile to point out that using these representations, it is easy to see that:
Here are some more good examples:
Then, each pair should compare their intervals with other pairs. If two intervals overlap, then on the overlap both series should converge and they can be added together. The pairs should join together by adding their series to give what is unlikely to be a simple series. Then together they will be able to find the sum of this more complicated series on the overlapping interval.
As a simple example, one pair could be given the geometric series
which is not a geometric series. However it is the sum of two simple geometric series, each of which converge, so the limit should be:
Students should be asked to find the Taylor and Maclauren Series expansions for a variety of complicated functions using the formulas that:
The only difficulty in carrying this out is in obtaining a general formula for the n^th derivative of a function. Therefore the best questions lead them towards the answer in the following sort of way: