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.

E 9.1 Exercises