## 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).-
**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: -
**Do**the*Preview Activity for this section*(on WeBWorK if required by your teacher). -
**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: -
**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*)

- Plotting a sequence in Desmos (

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.