© 2000 M.
Flashman

Dec., 2005: This page now requires Internet Explorer 6+MathPlayer or Mozilla/Firefox/Netscape 7+.

**Preface**: Throughout this introduction to calculus we have
found
that estimates could be improved by some process determined by a
natural
number `n`. In our estimates for solving equations this number `n`
denoted
the number of steps we would follow in using either the bisection
method
or Newton's method. Later `n` was the number of steps we used in
Euler's
method to estimate the value of the solution to a differential
equation
with given initial (or boundary) condition. Later still we found
`n` controlling
the size of intervals and the quality of estimates for the value
of a definite
integral with the trapezoidal rule and Simpson's rule. The number
`n` also
was used in estimating the values of many measurable quantities in
applications,
such as area and volume, allowing us to connect real situations to
the
mathematical concepts of derivative and integral. Finally, in the
last
chapter we found `n` indicating the degree of the MacLaurin and
Taylor polynomials
and their remainder terms which gave estimates for values for
functions,
integrals and solutions to differential equations.

The mathematics of

One crucial question considers whether there is a tendency in the sequence for its members associated with larger natural numbers to approach a particular value called the limit of the sequence. There are actually two important and related questions:

(1) Is there a limit for a particular sequence?

For example, we are fortunate sometimes when we can use the Fundamental Theorem of Calculus to determine the precise value of the limit of numbers arising from Simpson's Rule for estimating a definite integral. Or in other situations we can use the Taylor Theorem to recognize that `1 + 1 + 1/2+ 1/6 + 1/{4!} + … + 1/{n!}` is approximately `e`. And we shouldn't forget that the sequences that arise from Newton's method for estimating solutions to equations (square roots, etc) frequently converge to numbers that are well described but known primarily by their estimates.

{`a_n : n = 0,1,2, …` } | {`b_k`}`k = 0,1,2, …` | {`c_k`}`k = 1,2,3, …` |

{`x_ n`}`n = 0,1,2, …` | {`y_n`} `n = 0,1,2, …` | {`s_n`} `n = 0,1,2, …` |

A typical defining statement for a sequence is `a_n
= f(n)` where `f` is some function defined at least on the
natural numbers.

For example `a_n = n^2`, `b_n
= 1/{n+1}`; `x_k = 1/{k!}`; and `s_n =
sum_{k=0}^{k=n}1/{k!}`.

It is also possible to define a term of a sequence using
previous
terms, such as `r_0=2` and for `k>=0, r_{k+1} = {r_k^2 +
2}/{2r_k}`
[Coming from Newton's method to approximate the square root of 2.]
Another example
of this sort is the sequence defined by `f_0=1, f_1=1` and
`f_{k+2}=f_k + f_{k+1}`, creating the sequence
`1,1,2,3,5,8,13,....` often described as (Leonardo Pisano)
Fibonacci's
sequence (1170 - 1250) .

An even more general example would allow the sequence to have a
real
variable in its definition, such as the sequence

**Visualizing sequences**: There are many ways to think about
real
number sequences and each way leads to a visualization of the
sequence
that can assist in making the sequence and its properties more
apparent.

First, a real number sequence is **a set of real numbers
together with
an order** for these numbers. With this in mind we can ** display
the
sequence as a collection of points on a real number line**.
The
order of the points can be seen by labeling the points with the
subscripts
that indicate there relative

order, such as `a_1, a_2, a_3, …` .

Example. X.I.A

A visualization of the sequence {`1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7,
1/8,
1/9, 1/10, ..., 1/k ,...`}

Another way to indicate the order is to draw an arrow from one point to the next on the line, so we see a succession of arrows above the number line each showing which number is next in the sequence.

A second and important way to treat a sequence is as

One issue that is very important in the study of sequences is
whether
the values of the sequence
{`a_n`} `n=0,1,2,_{ …`} approach a limiting number,
`L`, when `n` is very large.
If this is the case we write that* *`a_n ->
L` as `n ->oo` or `lim_{n->oo}a_n = L` and we say, "The
sequence
an converges to L as n goes to infinity."

**Examples**. Let `a_n = n^2` ,`b_n
=1/{n+1}`, `c_n = {4n^2} /{7 n^2 - 5n
- 2}`, and `S_n = cos(1/{1 + n^2 })`.

Discussion: These sequences might also be described directly
as { `n^2` }`n = 0,1,2, …`; { `1/{n+1}` }`n
= 0,1,2,…`; `{4 n^2} /{7 n2
- 5 n - 2}`

`n = 0,1,2, …`; and `cos(1/{1 + n^2}) n
= 0,1,2,` …. Table X.A.1 shows some of
the
initial values for the sequences while the figures below show in
part the mapping diagrams and graphs for initial values of
the latter three
sequences.

Without being rigorous at this stage, it should make sense that
there
are limits for all but the first of these sequences. In particular

`lim_{n->oo}1/{n+1} =
0 ; lim _{n->oo}{4n^2}
/{7 n^2-5n-2}= lim _{n->oo}{4}
/{7 - 5/n - 2/{n^2}}= 4/7`; and `lim _{n->oo}cos(1/{1 +
n^2 }) = 1`.

**Examples from Euler's method and Riemann sum integral
approximations.**

**Example.** A sequence that converges to `e`. Recall that in
considering
the differential equation `y'=y` with `y(0)=1` we estimated the
value of `y(1)`
using Euler's method in `n` steps to be `(1+1/n)^
n`. With `n` larger
we obtain better estimates for `y(1)=e`. Thus we can conclude that
as `n->oo, (1+1/n)^ n -> e`.

**Example**. A sequence that converges to `ln(2)`. In
considering the
differential equation `y'=1/x` with `y(1)=0`, using Euler's method
in
n steps, we estimated

`y(2)~~ 1/n [1 +
n/{n+1} + n/{n+2} + …. + n/{2n-1}] = 1/n + 1/{n+1} + … + 1/{2n-1}
= S_n`.
Thus as `n->oo, S_n ->
y(2) = ln(2)`. Notice that this sequence is also the sequence of
left hand
endpoint Riemann sum estimates of the definite integral `int_1^2
1/x
dx`.

**Example**. A sequence that converges to `1/2`. For a simpler
example
we can consider `S_n = 1/n [ 1/n + 2/n + …. + {n-1}/n]`, the left
hand Riemann sum estimate for the definite integral `int_0^1 x
dx
=
1/2`* *. Thus as `n->oo, S_ n->
1/2`.

**Examples from Taylor approximations.**

**Example**. Another sequence that converges to e. Recall that
in
considering the differential equation `y' = y` with `y(0)=1` we
estimated the
value of `y(1)` using Taylor (Maclaurin) polynomials of degree `n`
evaluated
at `x=1`. If we let `S_n = P_n
(x) = 1 + 1 + 1/2 + 1/{3!}
+ 1/{4!} + ... + 1/{n!}`, then with n larger we obtain
better estimates
for `y(1)=e`. [In fact we actually have a measure of the
difference between
`e` and `S_n`_{
}given by `R_n = e - S_n` as in the
work on Taylor Theory in chapter IX.] Thus we can conclude that as
`n->oo,
S_n -> e`. Furthermore, the same
theory explains why for each *x,* the sequence

`S_n = P_n
(x) = 1 + x + {x^2}/2 + {x^3}/{3!}
+ {x^4}/{4!} + ... + {x^n}/{n!}` converges
to `e^x` as `n->oo`..

**Example**. A sequence that converges to `ln(1.1)`. Recall
that in
considering the differential equation `y' = 1/x` with `y(1)=0` we
estimated
the value of `y(1.1)` using Taylor polynomials of degree `n` about
`x=1`
evaluated at `x=1.1`. If we let

`R_n = ln(1.1) - S_n` as in the work on Taylor Theory in chapter IX.] Thus we can conclude that as `n->oo`, `S_n -> ln(1.1)`.

**Exercises X.A.**

For each of the following sequences, (a) list the first five values,(b) draw a mapping diagram for the first five values, (c) draw a graph for the first five values, and (d) find the limit as `n->oo` when possible.

1. `((5n^2 + 3n -2)/(2n^2 + 3n -1)) _ {{n = 0,1,2 ...}}`.

2. ` sin( (3n -2)/(2n^2 + 3n -1)) _ {{n = 0,1,2 ...}}`.

3.`((4n + 2)/(3n^2 + 2n -1)) _ {{n = 0,1,2 ...}}`.

4. `((n^3 + 1)/(3n^2 + 2n)) _ {{n = 1,2 ...}}`.

5. (-1)^{n}_{{n = 0,1,2 ...}}.

6. (-1)^{n}/(n^{2} + 1) _{{n = 0,1,2 ...}}.

7. (2/3)^{n} _{{n = 0,1,2 ...}}.

8. (-2/3)^{n}_{{n = 0,1,2 ...}}.

9. e^{1/(n+1)} _{{n = 0,1,2 ...}}.

10. `tan^(-1)(1 - 1/(n+1)) _ {{n = 0,1,2 ...}}`.