A discovery-based
Introduction to Infinite Sequences

First Definition An infinite sequence (or just "sequence") is an infinite listing of numbers, like

1, 2, 3, 4, 5, 6, .... (etc)

or

1, 1/2, 1/3, 1/4, .... (etc)

or

0, 0, 0, 0, 0, .... (etc)

or

8.17, 1, 2, 3, 3, 3, 3, 3, .... (3's forever)

or

3, 1, 4, 1, 5, 9, 2, ....

Create your own infinite sequence:

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How about another one:

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Some notation. Just as we sometimes like to use constants like "c" and "K" instead of actual numbers like "3" and "8.64", there is a corresponding notation for sequences:

a1, a2, a3, a4, ...

or in braces:

{a1, a2, a3, a4, ...}

or more concisely, just:

{an}


Here "n" is just an "index" or "dummy variable". For example,

the sequence {an}

is the same as the sequence {ak}



In particular,

the sequence {1/n}

is the same as the sequence {1/k}

Note that both of these represent (equal)
one of the sequences above, namely:

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Convergence
We say that "the sequence {an} converges to L as n tends to infinity", and write

an ® L  as  n®

or 

 Lim  an = L 
n®       

For example, the limit of the constant sequence {c, c, c, c, ...} is c.

Given a sequence {an}, there are several possibilities as to its convergence behavior:

  1. The sequence {an} may converge to a number L.

  2. The sequence may diverge in one of three ways:

    1. The sequence may tend to +
      (We sometimes say "goes to infinity" for this type of divergence)

    2. The sequence may tend to -
      (We sometimes say "goes to negative infinity" for this type of divergence)

    3. The limit might not exist at all.

Which type is the sequence {1/k}? __________ The sequence (-1)n/n? __________ The sequence (-1)k? __________.

Now write at least two (can you come up with three or more?) sequences whose convergence is of each of the types listed above (you may use some of the sequences listed on the first page).

Sequences of type 1:




Sequences of type 2A:




Sequences of type 2B:




Sequences of type 2C:




Another definition
A sequence {an} can also be thought of as a function f whose domain is the counting numbers {1,2,3,...} and whose range is the real numbers: f(n) = 1/n for example.

The graph of such a function is a sequence (infinite of course!) of points. On the axes below, put in the graphs of the three infinite sequences given by f(n) = 1/n, g(n)= n, and h(n)= 2-(1/n). Be sure to label the units on both axes. 

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Speaking informally but being as precise as you can, describe how the graphs above relate to convergence. What visually/geometrically is key to convergence?

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Now graph f(n)=(-1)n/n and g(n) = (-1)n (in different colors): 
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What do these graphs tell you about convergence? Why?
Be as mathematically precise and convincing as you can. 
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____________________________________________________________
 "Functional Fact": If an=f(n) for every integer n, where f(x) is a function defined on the real line, then the limit as n ® of an equals the limit as x ® of f(x). (For example, since the limit as x®infinity of 1/x is zero for the function f(x)=1/x, this theorem promises that the limit as n®infinity of 1/n must also be 0. But not every sequence of an terms will be just f(n) for an obvious formula/function f(x))

For each of the following functions

  1. Find (or guess) the limit of the corresponding sequence, if it exists, algebraically (by analyzing the formula, but without using the "Functional Fact" directly).
  2. Graph the function below
  3. Interpret your answer to a. in light of b.

1. f(n) = n/(n+1)   Experimentally:________________________________________ Diverges? Or else converges to?__________.
2. g(k) = sin(k)/k   Experimentally:________________________________________ Diverges? Or else converges to?__________.

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Part c.:


Algebra Facts

If an® L and bn® M, then Technical Aside
To prove these, one uses the precise N-epsilon definition of limits of sequences, and the fact the each of these algebraic operations is "continuous". For example, addition is continuous in the sense that, if X is close to Y and A is close to B, then X+A is close to Y+B, and that furthermore, we can make X+A be as close as anyone demands, so long as we're allowed to make A and X, and B and Y, respectively, sufficiently close to each other.

Dealing with Exponentials
Henceforth, "the limit of" will mean "the limit as <the variable> approaches " unless stated otherwise.
Fact: If the limit of {an} is L (and L>0), then by letting bn = ln(an) for every n, we get a sequence such that the limit of {bn} is ln(L). This means conversely that if we don't know the limit of {an}, but find that the limit of {bn} is M, then the limit of {an} is eM.

For example, suppose we wish to find the limit of {an} where an = n(1/n)

If bn = ln(an) for each n, then by the properties of logarithms, bn = (ln(n))/n.

Using L'Hopital's rule on f(x) = (ln(x))/x we see that the limit of f(x) is _______. Hence by the result "____________________" above, the limit of {bn} is ___________. Hence by the fact about exponentials, the limit of the original sequence {an} is __________.

B R E A K (Optional: Sequences Worksheet and/or presentations)
...then continue with Extra Topics...
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