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.

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?

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.

Example. X.I.A (continued). Let `a_n = 1/n` for `n = 1,2,3,...` . In the mapping diagram the index values, `n`, will be located on the source line while the actual values of the sequence, `a_n`, will be corresponding points on the target line. An arrow is drawn from the index point in the source to the corresponding value point of the sequence on the target. In fact the target line will look essentially like our first visualization without arrows to indicate the order. The order in the mapping diagram is indicated by the usual order of the indices on the source line.

In the graph of a sequence we see a collection of points in the plane with ordered pairs of real numbers for coordinates. The first coordinate is the index of the number in the sequence, `n`, while the second coordinate is the actual value of the sequence, `a_n`. The order in the graph is seen as the fact that values with larger indices are represented by points further to the right on the graph.

One issue that is very important in the study of sequences is whether the values of the sequence {`a_n`} `n=0,1,2, <sub>…`</sub> 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`.

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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`.

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

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/(n² + 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 ...}}`.