Sequences

Section Sequences

Definition 6.1.

A sequence is a function with domain a subset of the natural numbers and range a subset of the real numbers.

Example 6.2.

We will denote our sequences by listing elements of the range of the sequence.

\(\dsp{\frac{1}{2} \;, \frac{1}{4} \; , \frac{1}{8}}\; , \dots\) denotes the range of the sequence \(\dsp f(n) = \frac{1}{2^n}, \; n = 1,2,3,\dots\text{.}\)
\(a_1, a_2, a_3, \dots\) denotes the range of the sequence \(f(n) = a_n, \; n = 1,2,3,\dots\text{.}\)
\(\dsp \{x_n \}_{n=1}^{\infty}\) denotes the range of the sequence \(f(n) = x_n, \; n = 1,2,3,\dots\text{.}\)
\(\dsp \{x_n=\frac{n}{n+1} \}_{n=1}^{\infty}\) denotes the range of the sequence \(\dsp f(n) = \frac{n}{n+1}, \; n = 1,2,3,\dots\text{.}\)

We mathematicians abuse the definition of function by referring to \(y=x^3\) as a function, when in fact the function is the collection of ordered pairs generated by this equation. In a similar abuse, while \(\{x_n \}_{n=1}^{\infty}\) or \(x_1, x_2, x_3, \dots\) represents the elements of the range of the sequence, we refer to it as though it were the sequence itself.

Definition 6.3.

The sequence \(\{x_n \}_{n=1}^{\infty}\) is said to be bounded above if there is number \(M\) such that \(x_n \le M\) for all \(n \ge 1\text{.}\) If such a number exists, then \(M\) is called an upper bound for the sequence. Bounded below and lower bound are defined in the analogous way. The sequence is bounded if it is bounded above and bounded below.

Problem 6.4.

Show that the sequence, \(\dsp \{x_n = \frac{1}{n} \}_{n=1}^{\infty}\) is bounded above and below.

Definition 6.5.

The sequence \(\{x_n \}_{n=1}^{\infty}\) is increasing (often called strictly increasing) if \(x_n \lt x_{n+1}\) for all natural numbers \(n\text{.}\) The sequence \(\{x_n \}_{n=1}^{\infty}\) is non-decreasing if \(x_n \le x_{n+1}\) for all natural numbers \(n\text{.}\) Decreasing and non-increasing are defined similarly. A sequence is said to be monotonic if it is either increasing, decreasing, non-increasing, or non-decreasing.

Definition 6.6.

The factorial function is defined for all non-negative integers as follows:

\(0!=1\)
\(n! = n \cdot (n-1)!\) for \(n=1,2,3,\dots\text{.}\)

The first few factorials are:

\begin{equation*} 0! = 1, \ \ \ 1! = 1, \ \ \ 2! = 2 \cdot 1 = 2, \ \ \ 3! = 3 \cdot 2 \cdot 1 = 6, \ \ \ 4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24, \dots \end{equation*}

Problem 6.7.

For each of the following, prove that it is monotonic or show why it is not monotonic.

\(\dsp \{\frac{n+3}{4n+2} \}_{n=1}^{\infty}\)
\(\dsp \{ \sin({\frac{n \pi}{3}}) \}_{n=1}^{\infty}\)
\(\dsp \{\frac{(n-1)!}{2^{n-1}} \}_{n=1}^{\infty}\)

Definition 6.8.

Given the sequence, \(\{x_n \}_{n=1}^{\infty},\) the number \(G\) is called the greatest lower bound of the sequence if \(G\) is a lower bound for the sequence and no other lower bound is larger than \(G.\) The number \(L\) is called the least upper bound if \(L\) is an upper bound and no other upper bound of the sequence is less than \(L.\)

Problem 6.9.

Find the least upper bound and the greatest lower bound for these sequences.

\(\dsp \{ \frac{n}{n+1} \}_{n=1}^\infty\)
\(\dsp \{ \frac{2^n}{n!} \}_{n=2}^\infty\)
\(\dsp \{ \frac{2n+1}{n} \}_{n=2}^\infty\)

Problem 6.10.

For each sequence, write out the first five terms. Is there is a number that the values of \(x_n\) approach as \(n \rightarrow \infty\text{?}\) If so, what is it. If not, why not?

\(\dsp\{ x_n = \frac{1}{n} - \frac{1}{n-1}\}\) for \(n=2,3,4, \dots\)
\(\dsp\{ x_n = (-1)^n (2-\frac{1}{n})\}\) for \(n=1,2,3,\dots\)
\(\dsp\{\frac{n^2}{n!}\}_{n=5}^\infty\)

Definition 6.11.

Given a sequence, \(\{x_n \}_{n=k}^{\infty}\text{,}\) we say that the sequence converges to the number \(L\) if for every positive number \(\epsilon\text{,}\) there is a natural number \(N\) so that for all \(n \ge N,\) we have \(|x_n - L| \lt \epsilon.\)

If \(\{x_n \}_{n=k}^{\infty}\) converges to \(L,\) we may write “\(\displaystyle{\lim_{n \to \infty} x_n = L}\)” or “\(\displaystyle{x_n \to L}\) as \(n \to \infty.''\) We call a sequence convergent if there is a number to which it converges, and divergent otherwise. A divergent sequence might tend to \(\pm \infty\) or might ``oscillate.” In either case, there is no single number that the range of the sequence approaches.

Problem 6.12.

Consider the sequence, \(\dsp \frac{5n+4}{3n+1}.\) Let \(\epsilon = 0.00002\) and find the smallest natural number \(N\) so that \(\dsp |\frac{5n+4}{3n+1} - \frac{5}{3}| \lt \epsilon\) for every \(n \ge N\text{.}\) Repeat for \(\epsilon = 0.00001\text{.}\)

Problem 6.13.

Consider the sequence, \(\dsp \frac{n^2}{n^2+1}.\) Let \(\epsilon = .0004\) and find the smallest natural number \(N\) such that \(\dsp |\frac{n^2}{n^2+1} - {1}| \lt \epsilon\) for all \(n \ge N\text{.}\) Repeat for \(\epsilon = .0002\text{.}\)

Problem 6.14.

Let \(\epsilon\) represent a positive number less than one. Find the smallest natural number \(N\) such that \(\dsp |\frac{5n+4}{3n+1} - \frac{5}{3}| \lt \epsilon\) for every \(n \ge N\text{.}\) The number \(N\) will depend on \(\epsilon\) since a smaller value for \(\epsilon\) may result in a larger value for \(N.\)

Problem 6.15.

Let \(\{x_n\}_{n=1}^{\infty}\) be the sequence so that \(x_n = 0\) if \(n\) is odd and \(x_n=1\) if \(n\) is even.

Show that the sequence is bounded.
Show that the sequence does not converge to \(0\text{.}\)
What would we need to prove in order to show that the sequence is divergent?

Problem 6.16.

Let \(\epsilon\) represent a positive number less than one. Find the smallest positive integer \(N\) such that \(\dsp |\frac{n^2}{n^2+1} - {1}| \lt \epsilon\) for all \(n \ge N\text{.}\)

Problem 6.17.

Show that if \(\{ a_n \}_{n=1}^\infty\) is a convergent sequence then it is a bounded sequence.

Definition 6.18.

The sequence \(\{x_n \}_{n=k}^{\infty}\) is called a Cauchy sequence if for every \(\epsilon > 0\text{,}\) there is a natural number \(N\) so that \(|x_n - x_m| \lt \epsilon\) for all \(m, n \ge N\text{.}\)

Problem 6.19.

Show that if \(a\) and \(b\) are numbers then \(|a+b| \le |a| + |b|\text{.}\)

Problem 6.20.

Show that if \(\{ a_n \}_{n=1}^\infty\) is a convergent sequence then it is a Cauchy sequence.

It is also true that if \(\{ a_n \}_{n=1}^\infty\) is a Cauchy sequence then it is a convergent sequence. That's a bit more interesting problem than we have time to explore — take Analysis.

Definition 6.21.

The sequence \(\{x_n \}_{n=k}^{\infty}\) is called a C sequence if for every \(\epsilon > 0\text{,}\) there is natural number \(N\) such that \(|x_N - x_{n}| \lt \epsilon\) for all \(n \ge N\text{.}\)

Problem 6.22.

Two parts.

Suppose that \(\{ a_n \}_{n=1}^\infty\) is a C sequence. Is it a Cauchy sequence?
Suppose that \(\{ a_n \}_{n=1}^\infty\) is a Cauchy sequence. Is it a C sequence?

An axiom is a statement that we use without proof. All of mathematics is built on axioms since one must make certain assumptions just to get started. There are entire branches of mathematics devoted to determining what of the mathematics that we use now would still be true if we changed the underlying axiom system that mathematicians (more or less) universally agree on.

The Completeness Axiom says that if we have a set of numbers and there is some number that is greater than every number in our set, then there must be a smallest number that is greater than or equal to every number in the set.

Axiom 6.23.

Completeness Axiom If a non-empty set \(S\) of real numbers has an upper bound, then \(S\) has a least upper bound.

Theorem 6.24.

Monotonic Convergence Theorem. Every bounded monotonic sequence is convergent.

Problem 6.25.

Show that \(\dsp \{ \frac{2^n}{n!} \}_{n=1}^{\infty}\) is convergent.

The next theorem formalizes the relationship between sequences and limits of functions.

Theorem 6.26.

Let \(\{x_n \}_{n=1}^{\infty}\) be a sequence and \(f\) be a function defined for all \(x \ge 1\) such that \(f(n)=x_n\) for all \(n=1,2,3.\dots\text{.}\) The only difference between the function \(f\) and the function \(x\) (remember, a sequence is a function) is that the domain of \(f\) includes real numbers and the domain of \(x\) is only natural numbers.

If \(\displaystyle{\lim_{x \to \infty} f(x) = L}\text{,}\) then \(\displaystyle{\lim_{n \to \infty} x_n = L}\text{.}\)
If \(\displaystyle{\lim_{x \to \infty} f(x) = \pm \infty}\text{,}\) then \(\displaystyle{\lim_{n \to \infty} x_n = \pm \infty}\text{.}\)
If \(\displaystyle{\lim_{n \to \infty} x_n = L}\) and \(\displaystyle{\lim_{n \to \infty} y_n = M}\text{,}\) then
1. \(\displaystyle{\lim_{n \to \infty} (x_n \pm y_n) = L \pm M}\text{;}\)
2. \(\displaystyle{\lim_{n \to \infty} (x_n \cdot y_n) = LM}\text{;}\)
3. \(\displaystyle{\lim_{n \to \infty} \frac{x_n}{y_n} = \frac{L}{M}}\) for \(M \ne 0\text{.}\)

Problem 6.27.

Consider the sequence \(\dsp x_n = \frac{1}{n}\) for all \(n=1,2,3,\dots.\) Sketch and define a function, \(f\text{,}\) so that \(f\) agrees with \(x_n\) at all the natural numbers, but \(\dsp \lim_{x \to \infty} f(x) \ne 0.\)

Problem 6.28.

Evaluate each of the following limits, whenever it exists.

\(\displaystyle{\lim_{n \to \infty} \frac{n^2 + 2n + 3}{3n^2 + 4n -5}}\)
\(\displaystyle{\lim_{n \to \infty} \frac{6n + 100}{n^3 + 4}}\)
\(\displaystyle{\lim_{n \to \infty} \frac{\log_3 (n)}{n}}\)
\(\displaystyle{\lim_{n \to \infty} (1+\frac{3}{n})^n}\)