9.1 | Sequences

A sequence can be thought of as an ordered list of objects (numbers, functions, etc…):
$$a_1, a_2, a_3, a_4, \ldots, a_n, \ldots$$
where $$a_1$$ represents the first term, $$a_2$$ represents the second term, … , $$a_n$$ represents the $$n^{th}$$ term, $$a_{n+1}$$ represents the $$(n+1)^{th}$$ term, etc… Three common ways to represent a sequence:

• $$\{ a_1, a_2, a_3, \ldots \}$$
• $$\{ a_n \}$$
• $$\{ a_n \}_{n=1}^{\infty}$$
A sequence $$\{ a_n\}$$ has limit $$L$$ and we write
$$\lim_{n \rightarrow \infty} a_n = L$$
if we can make the terms $$a_n$$ as close to $$L$$ as we like by taking $$n$$ sufficiently large. If $$\lim_{n \rightarrow \infty} a_n$$ exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).
If
$$\lim_{x \rightarrow \infty} f(x) = L \text{ and } f(n) = a_n$$
when $$n$$ is an integer, then
$$\lim_{n \rightarrow \infty} a_n = L.$$
If $$\{ a_n \}$$ and $$\{ b_n \}$$ are convergent sequences and $$c$$ is a constant, then

• Sum/Difference:
$$\lim_{n \rightarrow \infty} (a_n \pm b_n) = \lim_{n \rightarrow \infty} a_n \pm \lim_{n \rightarrow \infty} b_n$$
• Constant Multiplier:
$$\lim_{n \rightarrow \infty} c a_n = c \lim_{n \rightarrow \infty} a_n$$
• Constant:
$$\lim_{n \rightarrow \infty} c = c$$
• Product:
$$\lim_{n \rightarrow \infty} (a_n b_n) = \lim_{n \rightarrow \infty} a_n \cdot \lim_{n \rightarrow \infty} b_n$$
• Quotient:
$$\displaystyle \lim_{n \rightarrow \infty} \frac{a_n}{b_n} = \frac{\displaystyle \lim_{n \rightarrow \infty}a_n}{\displaystyle \lim_{n \rightarrow \infty}b_n},$$
provided $$\displaystyle \lim_{n \rightarrow \infty} b_n \not = 0$$
• Power:
$$\lim_{n \rightarrow \infty} a_n^p = \left[\lim_{n \rightarrow \infty} a_n \right]^p,$$
provided $$p>0$$ and $$a_n > 0$$
The Squeeze Theorem: If $$a_n \leq b_n \leq c_n$$ for $$n \geq n_0$$ and
$$\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} c_n = L,$$
then $$\displaystyle \lim_{n \rightarrow \infty} b_n = L$$.
If $$\displaystyle \lim_{n \rightarrow \infty} |a_n| = 0$$, then $$\displaystyle \lim_{n \rightarrow \infty} a_n = 0$$.
If $$\displaystyle \lim_{n \rightarrow \infty} a_n = L$$ and the function $$f$$ is continuous at $$L$$, then
$$\lim_{n \rightarrow \infty} f(a_n) = f(L)$$
The sequence $$\{r^n\}$$ is convergent if $$-1 < r \leq 1$$ and divergent for all other values of $$r$$. That is, $$\lim_{n \rightarrow \infty} r^n = \begin{cases} 0, & \; \text{ if } -1 < r <1 \\ 1, & \; \text{ if } r = 1 \\ \text{divergent}, & \; \text{otherwise} \end{cases}$$
A sequence $$\{ a_n \}$$ is called increasing if $$a_n < a_{n+1}$$ for all $$n \geq 1$$, that is, $$a_1 < a_2 < a_3 < \cdots.$$ It is called decreasing if $$a_n > a_{n+1}$$ for all $$n \geq 1$$. A sequence is monotonic if it is either increasing or decreasing.
A sequence $$\{ a_n \}$$ is bounded above if there is a number $$M$$ such that
$$a_n \leq M \; \; \text{ for all } n \geq 1$$
It is bounded below if there is a number $$m$$ such that
$$m \leq a_n \; \; \text{ for all } n \geq 1$$
If it is bounded above and below, then $$\{ a_n \}$$ is a bounded sequences.
Monotonic Convergence Theorem for Sequences: Every bounded, monotonic sequence is convergent.