Sequences Review

We will now review some of the recent content regarding sequences.

• Recall from the Sequences page that a Sequence of real numbers $\{ a_n \}$ is an ordered list of numbers called Terms of the sequence, where $a_j$ represents the $j^{\mathrm{th}}$ term. Alternatively we can view a sequence of real numbers as a function from the natural numbers to the real numbers, that is, for each $n \in \mathbb{N}$ we assign a term value $a_n \in \mathbb{R}$.

• We said that a sequence is Bounded between the numbers $m$ and $M$, $m ≤ M$ if for all $n \in \mathbb{N}$ we have that $m ≤ a_n ≤ M$. A sequence is said to be Bounded Above if for all $n \in \mathbb{N}$ we have that $a_n ≤ M$, and a sequence is said to be Bounded Below if for all $n \in \mathbb{N}$ we have that $m ≤ a_n$.

• Furthermore, a sequence is said to be Strictly Increasing if for all $n \in \mathbb{N}$ we have that $a_n < a_{n+1}$. A sequence is said to be Strictly Decreasing if for all $n \in \mathbb{N}$ we have that $a_n > a_{n+1}$. A sequence that is either strictly increasing or strictly decreasing is said to be Strictly Monotonic. A sequence is said to be Ultimately Increasing if there exists an $N \in \mathbb{N}$ such that if $n ≥ N$ then $a_n < a_{n+1}$. A sequence is said to be Ultimately Decreasing if there exists an $N \in \mathbb{N}$ such that if $n ≥ N$ then $a_n > a_{n+1}$. A sequence that is ultimately increasing or ultimately decreasing is said to be Ultimately Monotonic.

• Additionally, a sequence is said to be Positive if for all $n \in \mathbb{N}$ we have that $a_n > 0$, and a sequence is said to be Negative if for all $n \in \mathbb{N}$ we have that $a_n < 0$. A sequence is said to be Ultimately Positive if there exists an $N \in \mathbb{N}$ such that if $n ≥ N$ then $a_n > 0$ and a sequence is said to be Ultimately Negative if there exists an $N \in \mathbb{N}$ such that if $n ≥ N$ then $a_n < 0$.

• On the Limit of a Sequence page we said that a sequence $\{ a_n \}$ Converges to the real number $L$ written $\lim_{n \to \infty} a_n = L$ if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - L \mid < \epsilon$. If a sequence does not converge then it is said to Diverge.

• We also looked at an important theorem which says that if $\{ a_n \}$ is a sequence and $f(x)$ is a function then if $f(n) = a_n$ and $\lim_{n \to \infty} a_n = L$ then $\lim_{n \to \infty} f(n) = L$.

• We also noted that if $\lim_{n \to \infty} a_n = L$ and a function $f$ is continuous at $L$ then $\lim_{n \to \infty} f(a_n) = f(L)$.

• We then looked at a bunch of limit laws for sequences - all of which are analogous to the limit laws for functions.

Addition of Convergent Sequences	If $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$ then $\lim_{n \to \infty} (a_n + b_n) = A + B$.
Difference of Convergent Sequences	If $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$ then $\lim_{n \to \infty} (a_n - b_n) = A - B$.
Product Law of Convergent Sequences	If $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$ then $\lim_{n \to \infty} (a_nb_n) = AB$.
Quotient Law of Convergent Sequences	If $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B \neq 0$ then $\lim_{n \to \infty} \left ( \frac{a_n}{b_n} \right ) = \frac{A}{B}$.
Constant Multiple Law of Convergent Sequences	If $\lim_{n \to \infty} a_n = A$ and $k \in \mathbb{R}$ then $\lim_{n \to \infty} ka_n = kA$.
Power Law of Convergent Sequences	If $\lim_{n \to \infty} a_n = A$ and if $k$ is a nonnegative integer then $\lim_{n \to \infty} a_n^k = A^k$.
Squeeze Theorem for Convergent Sequences	If $a_n ≤ b_n ≤ c_n$ ultimately, and $\lim_{n \to \infty} a_n = L = \lim_{n \to \infty} c_n$ then $\lim_{n \to \infty} b_n = L$.

• The proofs of all of these theorems are given on the Limit Sum/Difference Laws for Convergent Sequences, Limit Product/Quotient Laws for Convergent Sequences, Limit Constant Multiple/Power Laws for Convergent Sequences, and The Squeeze Theorem for Convergent Sequences pages.

• On the Uniqueness of a Convergent Sequence's Limit page, we saw further that a convergent sequence has a unique limit.

• On the Proof that Convergent Sequences are Bounded page we also saw that if a sequence is convergent then is must be bounded. However, do note that the converse of this theorem is not necessarily true - that is, a bounded sequence need not be convergent. A prime example of a bounded sequence that is not convergent is the sequence $\{ (-1)^n \} = \{ -1, 1, -1, 1, -1, ... \}$ which alternates between $-1$ for odd terms and $1$ for even terms.

• We also noted from the The Monotonic Sequence Theorem for Convergence that if a sequence of real numbers is bounded and monotonic then this sequence must converge.

• On the Comparison Theorems for Sequences we proved that if $a_n < b_n$ ultimately and if $\lim_{n \to \infty} a_n = \infty$ then $\lim_{n \to \infty} b_n = \infty$. Furthermore, if $a_n > b_n$ ultimately and and $\lim_{n \to \infty} a_n = -\infty$ then $\lim_{n \to \infty} b_n = -\infty$. More generally, if $a_n < b_n$ ultimately and both the sequences $\{ a_n \}$ and $\{ b_n \}$ converge to the limits $A$ and $B$ respectively, then $\lim_{n \to \infty} a_n = A < B = \lim_{n \to \infty} b_n$.

• On the Evaluating Limits of Sequences we noted that evaluating limits of sequences is much like evaluating limits of functions. For example, consider the evaluating the limit of the sequence $\left \{ \frac{n^2 + n + 1}{n^3 + 2n} \right \}$. We note that:

(1)

\begin{align} \quad \lim_{n \to \infty} \frac{n^2 + n + 1}{n^3 + 2n} = \lim_{n \to \infty} \frac{\frac{n^2}{n^3} + \frac{n}{n^3} + \frac{1}{n^3}}{\frac{n^3}{n^3} + \frac{2n}{n^3}} = \lim_{n \to \infty} \frac{\frac{1}{n} + \frac{1}{n^2} + \frac{1}{n^3}}{1 + \frac{2}{n^2}} = \frac{0}{1} = 0 \end{align}

• On the Evaluating Limits of Recursive Sequences we saw that we can often times evaluating limits of sequences that are defined recursively if we can show that the sequence is ultimately monotonic and bounded (and hence a limit exists by the Monotonic Sequence theorem).