Finite vs. Infinite Sequences | Free Discrete Mathematics Lesson (4 College Credits) | UPI Study

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Finite vs. Infinite Sequences

Discrete Mathematics · Chapter 3: Sequences and SeriesSection 3.2 · lesson 18 of 878 min · free with a free account

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In this lesson

A last term - or a long run to ask questions about.

  • Identify the properties every finite sequence has
  • Describe convergence and divergence informally
  • Define bounded and unbounded sequences
  • Prove a sequence is bounded

Transcript

Everything said in the video, section by section (1,143 words). Read along, or search it.

Sequences and Series

Finite vs. Infinite Sequences

Section three point one introduced sequences. This section draws the line between finite and infinite ones, precisely. We'll see what every finite sequence is guaranteed to have. Then we'll describe, informally, what an infinite sequence can do in the long run. And we'll define bounded sequences, and prove that one is bounded.

The Opening Problem

An Export, and a Live Feed

A smart thermostat's temperature sensor logs one reading every minute. If a technician exports the readings from one twenty-four-hour maintenance window, the result is a fixed list. Exactly one thousand four hundred forty readings. That's a finite sequence. The live feed the thermostat runs on has no such boundary. It's produced a reading every minute since installation, and will keep going, with no last term. Its natural model is an infinite sequence. The distinction isn't just bookkeeping. It changes what questions are even meaningful. A finite sequence has a definite largest value, smallest value, and number of terms. An infinite sequence raises new questions. Does it settle down near some value? Does it keep growing forever? Or does it bounce around without settling at all?

Definition 3.2.1

Finite Sequences

A finite sequence has a fixed number of terms, n, written a one through a n. Formally, it's a function whose domain is one through n, rather than all the natural numbers. Three properties follow at once. It has a well-defined length, n itself. It has a first term, and, unlike an infinite sequence, a genuine last term. And because it's just a finite list of real numbers, it always has a maximum term and a minimum term. Every finite list has a largest and a smallest entry, with no exceptions. None of that is available to an infinite sequence. With no last term, the largest value and the last value simply aren't meaningful questions.

Worked Example

6 Days of Test Temperatures

Here's a six-day equipment test, recording daily high temperatures in degrees Fahrenheit. Sixty-eight, seventy-one, seventy, seventy-four, sixty-nine, and seventy-three. Six values are listed, so the length is six. The first term is sixty-eight, and the last term is seventy-three. Scanning all six, the largest is seventy-four, the fourth term. And the smallest is sixty-eight, the first term. Every one of these five facts is available precisely because the sequence is finite.

Definition 3.2.2

Infinite Sequences: Converge or Diverge

An infinite sequence is defined for every natural number, with no last term. Formally, its domain is all of the natural numbers. Because it never stops producing terms, it makes sense to ask what the terms do as n grows without bound. This section treats that question informally, as intuition. Making arbitrarily close precise belongs to a calculus-level treatment of limits. A sequence converges if its terms get closer and closer to one fixed number, as n grows without bound. That number is called its limit. A sequence diverges if it doesn't converge. Either its terms grow, or shrink, without bound, or they never settle toward any single value.

In Practice

Training Loss as a Sequence

The same distinction shows up directly in machine learning. Gradient descent produces a sequence of loss values, one per training step. A well-tuned run produces losses that appear to approach some minimum value. It converges. A learning rate set too high can produce losses that grow without bound, or that swing between high and low without ever settling. Both are divergence. Here's a worked example, as intuition. Take a sub n equals one over n. The first terms are one, one half, about a third, a quarter, a fifth, and so on. Each is smaller than the one before, and they visibly crowd closer and closer to zero. That's the informal picture of convergence. One over n converges, and its limit is zero.

Figure 3.2.1 (1)

Converging, and Growing Without Bound

This figure plots the first fifteen terms of four sequences. Here are the first two. On the left, one over n. The terms fall toward the dashed line at zero, its limit. On the right, a sub n equals n. One, two, three, four, growing without any upper limit. It diverges by growing unboundedly. Its terms exceed any proposed ceiling, once n is large enough.

Figure 3.2.1 (2)

Oscillating, and Converging to 1

Here are the other two. On the left, minus one to the n. Minus one, one, minus one, one. It never grows large at all, yet it never settles toward one value either. It diverges by oscillating. That case is worth pausing on. A sequence can fail to converge, even while every term stays confined to a small range. On the right, n over n plus one. One half, two thirds, three quarters, creeping up toward the dashed line at one. It converges to one.

Definition 3.2.3

Bounded and Unbounded

That oscillating example raises a distinction worth making precise. Staying within a fixed range is a different property from settling toward one value. A sequence is bounded above if there's a real number M with every term at most M. It's bounded below if there's an m with every term at least m. It's bounded if it's both, and unbounded otherwise. Every finite sequence is automatically bounded. Its maximum and minimum serve directly as M and m. So boundedness is a genuine question only for infinite sequences.

Worked Proof

n/(n+1) Is Bounded

Claim. The sequence n over n plus one is bounded, for n at least one. Let n be arbitrary, with n at least one. Then n and n plus one are both positive, so their ratio is positive. Every term is greater than zero. Now compare with one. n is less than n plus one for every n, since their difference is exactly one. Dividing both sides by the positive quantity n plus one gives n over n plus one, less than one. Together, every term lies strictly between zero and one. So the sequence is bounded, with m equal to zero and M equal to one.

Table 3.2.1

2 Different Properties

This table classifies all four sequences by both properties. One over n is bounded, between zero and one, and it converges to zero. n is unbounded, and it diverges by growing without bound. Minus one to the n is bounded, since every term is minus one or one. But it diverges, by oscillating. And n over n plus one is bounded, and converges to one. So boundedness and convergence are different properties. An unbounded sequence always diverges. But being bounded, on its own, doesn't guarantee convergence.

Takeaways

Finite Facts, Long-Run Questions

To summarize. A finite sequence has a definite number of terms, a first and last term, and a maximum and minimum. So it's always bounded. An infinite sequence has no last term, and questions about its long-run behavior only make sense for infinite sequences. It converges if its terms settle toward one value, and diverges if they grow without bound, or never settle. And boundedness means the terms stay between fixed limits, which is a different property from convergence. Here's something to think about. Minus one to the n is bounded, but doesn't converge. Does boundedness ever guarantee convergence on its own? And going the other way, can an unbounded sequence still settle down in some looser sense?

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FAQ

Questions students ask us

About Finite vs. Infinite Sequences

What properties does every finite sequence have?
Every finite sequence has a definite length n, a first term, a last term, a maximum term, and a minimum term. Because it is just a finite list of real numbers, some entry is largest and some is smallest. For example, the six temperatures 68, 71, 70, 74, 69, 73 have length 6, maximum 74, and minimum 68.
What is the difference between a convergent and a divergent sequence?
A sequence converges if its terms get closer and closer to one fixed number, called its limit, as n grows without bound. It diverges if it does not converge, either because the terms grow or shrink without bound or because they never settle on one value. For example, 1/n converges to 0, while a_n = n diverges.
What does it mean for a sequence to be bounded or unbounded?
A sequence is bounded if there are real numbers m and M with m \le a_n \le M for every term. That means it is bounded both above and below. If it fails to be bounded on either side, it is unbounded. Every finite sequence is bounded, since its minimum and maximum can serve as m and M.
How do you prove that the sequence n/(n+1) is bounded?
Show that every term lies between two fixed numbers. For n \ge 1, both n and n+1 are positive, so \frac{n}{n+1} > 0. Also n < n+1, and dividing by the positive number n+1 gives \frac{n}{n+1} < 1. So 0 < \frac{n}{n+1} < 1 for all n, and the sequence is bounded with m=0 and M=1.

About the Discrete Mathematics course

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Discrete Mathematics (Math 230) carries a NCCRS credit recommendation for 4 semester credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
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