2.1: Limits of Sequences

Suppose you have a sequence $a_n$.

Previously we inspected the first few terms of a sequence.

But we can also look at what happens at "the end" of the sequence, also called the tail of a sequence.

The Long-Term Behavior of a Sequence

What happens to the terms of the sequence when $n$ becomes large?

$a_n = \dfrac{1}{n}$
$b_n = (-1)^n$

The symbol $\infty$ is not a number. Certainly not a real number.

This is why in interval notation you always see $(-\infty, 3)$ or $(0, \infty)$, but never $(0, \infty]$.

Mathematicians use the symbol $\infty$ to represent the idea that a quantity grows forever.

A sequence $\{a_n\}$ has the 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 $\displaystyle\lim_{n\rightarrow \infty}a_n$ exists, we say the sequence converges (or is convergent)
Otherwise, we say the sequence diverges (or is divergent)

Note: $L$ must be a number. Otherwise the sequence is divergent.

Find what the sequence $a_n = \dfrac{1}{n}$ converges to.

Find what the sequence $a_n = 1$ converges to.

Is the sequence $a_n = \sqrt{n}$ convergent or divergent?

Limit Laws for Sequences

Making a table can be unwieldy and time consuming. Let's see how limits interact with operations.

If $\{a_n\}, \{b_n\}$ are convergent sequences and $c$ is a constant, then

$\displaystyle\lim_{n\rightarrow \infty}(a_n + b_n) = \displaystyle\lim_{n\rightarrow \infty}a_n + \displaystyle\lim_{n\rightarrow \infty}b_n$
$\displaystyle\lim_{n\rightarrow \infty}(a_n - b_n) = \displaystyle\lim_{n\rightarrow \infty}a_n - \displaystyle\lim_{n\rightarrow \infty}b_n$
$\displaystyle\lim_{n\rightarrow \infty}ca_n = c\displaystyle\lim_{n\rightarrow \infty}a_n$
$\displaystyle\lim_{n\rightarrow \infty}(a_nb_n) = \displaystyle\lim_{n\rightarrow \infty}a_n \cdot \displaystyle\lim_{n\rightarrow \infty}b_n$
$\displaystyle\lim_{n\rightarrow \infty}\dfrac{a_n}{b_n} =\dfrac{\displaystyle\lim_{n\rightarrow \infty}a_n}{\displaystyle\lim_{n\rightarrow \infty}b_n} \qquad$ if $\displaystyle\lim_{n\rightarrow \infty}b_n \neq 0$
$\displaystyle\lim_{n\rightarrow \infty}c = c$
$\displaystyle\lim_{n\rightarrow \infty}(a_n)^p = \left[\displaystyle\lim_{n\rightarrow \infty}a_n\right]^p$ for any number $p > 0$

Law 1 + 2 says the limit can be distributed among terms.

Law 4 says the limit can be distributed among factors.

Find $\displaystyle\lim_{n\rightarrow \infty}\left[2 + \dfrac{2}{n}\right]$ using limit laws.

From the previous example and limit law 7, we can deduce that

For all $p > 0$, we have \[\lim_{n\rightarrow \infty} \dfrac{1}{n^p} = 0 \] if $\frac{1}{n^p}$ is defined.

For example, the following limits are all equal to zero by the theorem: \[\lim_{n\rightarrow \infty} \dfrac{1}{n^2} = \lim_{n\rightarrow \infty} \dfrac{1}{\sqrt[3]{n^2}} = \lim_{n\rightarrow \infty} \dfrac{1}{\sqrt{n^3}} = 0 \]

Find \[\lim_{n\rightarrow \infty}\dfrac{1 + 2n^2}{5 + 3n + 4n^2}\]

Find \[\lim_{n\rightarrow \infty}\dfrac{n^3 + n - 1}{n^4 + n^2 + 3n}\]

Find \[a_n = \dfrac{n^2}{\sqrt{n^3 + 4n}}\]

Geometric Sequences

A geometric sequence is a sequence of the form $b_n = ar^n$. The sequence looks like \[a, \ ar, \ ar^2, \ ar^3, \ \dots \] The value of $r$ tells us if these sequences converge or not.

\[\lim_{n\rightarrow \infty} r^n = \begin{cases} 0 & 0 < r < 1 \\ 1 & r = 1 \\ \infty & r > 1 \end{cases}\]

Note: We use $\displaystyle\lim_{n\rightarrow \infty} r^n = \infty$ to denote the case where the terms $r^n$ grows without bound.

$\infty$ is not a number. All it means is $r^n$ grows without bound.

Find what $a_n = 1 - (0.2)^n$ converges to.

Find \[\lim_{n\rightarrow \infty} \dfrac{2^n - 1}{6^n}\]

Find \[\lim_{n\rightarrow \infty} \dfrac{e^n + e^{-n}}{e^{2n} - 1}\]