Overview
In Differential Calculus, we have considered limits of functions that were defined over a continuous data set. For instance, we have used limits to compute the limiting behavior of a velocity function dependent on time using \(\lim_{t \to \infty} v(t)\). However, in the real world, many data sets are not continuous, but rather discrete. For instance, the velocity may have been recorded only every second, and not at every possible instant: In that case, \(v(t)\) is not defined on a whole interval of numbers for \(t\), but rather only at the time values \(t=1, 2, 3, \ldots\), giving the (output) values \(v(1),v(2),v(3),\dots\). In other words, we have a function defined only for positive integers, and not for all real numbers. Nevertheless, it is still relevant to compute the limiting behavior of the velocity function \(\lim_{n \to \infty} v(n)\), to study what happens to the velocity “in the long run”. Note that the use of the letter \(n\) (instead of \(x\)) reminds us that only integer values are allowed for the variable \(n\).
In this section, we consider functions defined over the set of positive integers, and we call the list of resulting output values a sequence. Properties of continuous functions such as their limits, convergence, and monotonicity (increasing or decreasing) are also relevant for sequences and we will investigate how we can analyze the behavior of a sequence using the tools of (differential) calculus.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Define what a sequence is and define a sequence using appropriate notation.
- Graphing a sequence and using the graph to deduce the behavior (increasing, decreasing, limit (convergence)) of a sequence.
- Use simple limits to determine if a sequence converges.
- Define what it means for a sequence to converge to a number \(L\).
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Use limits to determine if a sequence converges, especially in the case where indeterminate forms arise. Use techniques from Calculus I such as conjugates, factoring leading terms, the Squeeze Theorem, and l’Hospital’s Rule.
- Determine if a sequence is increasing, decreasing, and/or bounded.
- Find the general term of a sequence given as a list of numbers.
To prepare for class
- Read the beginning of section 8.1 in Active Calculus (until Activity 8.1.2).
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Watch the following videos (by GSVUmath), always keeping in mind that you can change the playback speed on Youtube, and pause/rewind as you need. You can also activate [more or less accurate] subtitles/closed-captions for many videos —- and you should always make yourself some notes about what you have learned or noticed [or not understood] in a video:
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Do the Preview Activity for this section (on WeBWorK if required by your teacher).
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Watch the following video (by Dr. Trefor Bazett) which explains the meaning of convergence for a sequence and recalls the Limit Laws and how they are used in practice:
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Watch the following two videos (by patrickJMT) which go through several hands-on examples of applying different techniques from (differential) calculus to determine the convergence of a sequence (note that the first video unfortunately cannot be embedded on this page, so you’ll have to click on the link and then come back):
https://www.youtube.com/watch?v=Kxh7yJC9Jr0
After class
- Do some experimentation with the following interactive applets:
- Plotting a sequence in Desmos (Gabriel Indurskis)
- Arithmetic Sequences (nnhsmath)
- Arithmetic Sequence Practice: Predicting future terms when the first 4 are shown (nnhsmath)
- Dynamic Sequences (beckykwarren)
I will usually not point this out anymore, but you should always do the following after class:
- Read and review the whole section in Active Calculus.
- Finish any Activities from the section which we haven’t finished in class.
- Do the “regular”, after-class WeBWorK assignment for this section. You will typically have about 1 week to finish each assignment, but you should try to finish it within the first 2 or 3 days after it opens, as you will otherwise be in danger of falling behind. Most assignments will be relatively short, so try to get this done as fast as you can.
