Objectives
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Definition of a sequence.
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Definition of the limit of a sequence.
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Definition of the Monotonic Sequence Theorem.
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Presentation of the Squeeze Theorem.
Section 1.3 Sequences
Definition 1.3.1.
A sequence or equivalently an infinite sequence is a function \(f\) from the positive integers, \(\mathbb{Z}^{+}\) to the reals. The following \((a_{1},a_{2},a_{3},\ldots )\) is a sequence and can more compactly be written \((a_{n})_{n=1}^{\infty}\text{.}\)
Example 1.3.2.
The following are examples of infinite sequences:
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\(\displaystyle (1,2,3,4,\ldots)\)
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\(\displaystyle \displaystyle\biggl(1,\frac{1}{2},\frac{1}{3}, \ldots \biggr)\)
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\(\displaystyle (0,1,0,-1,0,1,0,-1,0,\ldots)\)
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\(\displaystyle (1,1,2,3,5,8,13,21,\ldots)\)
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\(\displaystyle \displaystyle \biggl(1,\frac{1}{2},\frac{2}{3},\frac{3}{5},\frac{5}{8}, \ldots \biggr)\)
The definition states that a sequence is a function from the positive integers to the reals, it is often nice to write down the function. It isn’t always easy to do. For example, the function of the first three sequences in Example 1.3.2, are \(a_{n}=n\text{,}\) \(b_{n}=\frac{1}{n}\text{,}\) \(c_{n} = -\cos (n\pi/2)\text{.}\) The fourth and fifth ones are not easily written as a function, but instead can be easily written recursively. For example, the fourth sequence is
\begin{equation*}
a_{n+1}= a_{n}+a_{n-1}, \qquad a_{1}=a_{2}=1
\end{equation*}
generates what is called the Fibonacci sequence. One may notice the same numbers in the fifth sequence and that can be written:
\begin{equation*}
b_{n+1}= \frac{1}{1+b_{n}}\qquad b_{1} = 1
\end{equation*}
Despite the fact that we could not write down a formula for the function to generated sequences 4 and 5 in Example 1.3.2, since a function is a rule, some function actually exists.
Subsection 1.3.1 Limits of Sequences
As we will see or you probably already know, it is important to know if the sequence approaches a number eventually. This is called the limit of the sequence and
Definition 1.3.3.
The limit of the infinite sequence \(\{a_{n}\}\) is \(L\) if for every \(\epsilon \gt 0\text{,}\) there exists a number \(N\) such that
\begin{equation*}
|a_{n} - L| < \epsilon
\end{equation*}
for all \(n\geq N\text{.}\)
Definition 1.3.4.
The sequence \(\{a_{n}\}\) converges to a number \(L\) if
\begin{equation*}
\lim_{n \rightarrow \infty}a_{n} = L
\end{equation*}
or
\begin{equation*}
\lim_{n \rightarrow \infty}|a_{n} -L| = 0.
\end{equation*}
Subsection 1.3.2 Monotonic Sequences
One important type of sequence is called a monotonic sequence, which is a sequence that increases or decreases.
Definition 1.3.5. Monotonic Sequence.
A monotonically increasing sequence is a sequence \(a_{n}\text{,}\) such that \(a_{n+1}\geq a_{n}\) for all \(n\text{.}\) A monotonically decreasing sequence is a sequence \(b_{n}\text{,}\) such that \(b_{n+1}\leq b_{n}\) for all \(n\text{.}\) A sequence that is either monotonically increasing or decreasing is called a monotonic sequence.
Example 1.3.6.
The following are monotonic sequences:
The first sequence is fairly obviously decreasing. However, the second is a bit less clear. The second is increasing, and can be shown by examining the derivative. If \(b_{n}=f(n)\text{,}\) then
\begin{equation*}
f'(n) = \frac{n^{2}+4n+1}{(n^{2}+2n+3)^{2}}
\end{equation*}
and this is positive for all \(n \gt 0\text{,}\) so the sequence is increasing.
We will see below that monotonic sequences have some helpful properties, including that bounded ones converge.
Theorem 1.3.7. Monotonic Sequence Theorem.
Every bounded, monotonic sequence is convergent.
Example 1.3.8.
Find the limits of the sequences in Example 1.3.2.
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Since \(a_{n}=n\text{,}\) this sequence grows without bound so the limit does not exist.
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In this case, \(b_{n}=\frac{1}{n}\text{,}\) which has a limit of 0.
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It is fairly clear that this sequence cycles (repeats) the values \(0,-1,0,1\text{,}\) so the limit cannot exist.
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As seen above, each element of the sequence is the sum of the previous two, so the limit cannot exist.
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Note that this sequence can be written For \(b_{n+1}= \frac{1}{1+b_{n}}\text{,}\) with \(b_{1}=1\text{.}\) Let’s assume that the limit exists, then\begin{equation*} \lim_{n \rightarrow \infty}b_{n} = L. \end{equation*}Taking the limit of both sides of \(b_{n+1}= \frac{1}{1+b_{n}}\) leads to\begin{equation*} \lim_{n \rightarrow \infty}b_{n+1}= \lim_{n \rightarrow \infty}\frac{1}{1+b_{n}} \end{equation*}or\begin{equation*} L = \frac{1}{1+L} \end{equation*}And solving for \(L\) leads to \(L=(-1+\sqrt{5})/2\approx 0.618)\text{.}\) If you take a number of terms of the sequence, you’ll see that it appears to be converging to this. We’ll see in Section ????? why this actually converges.
Subsection 1.3.3 The Squeeze Theorem
Consider the sequence given by \(\displaystyle a_{n} = \frac{(-1)^{n}}{n}\text{.}\) The first few terms are
\begin{equation*}
(-1, \frac{1}{2}, - \frac{1}{3}, \frac{1}{4}, -\frac{1}{5}, \ldots )
\end{equation*}
This is actually relative difficult to prove that it converges even though it appears that the sequence is approaching 0. However, the squeeze theorem is quite helpful:
Theorem 1.3.9. Squeeze Theorem.
Let \(a_{n}\) be a sequence that satisfies \(b_{n} \leq a_{n} \leq c_{n}\) for all \(n\text{,}\) where
\begin{equation*}
\lim_{n \rightarrow \infty}b_{n} = \lim_{n \rightarrow \infty}c_{n} = L
\end{equation*}
then
\begin{equation*}
\lim_{n \rightarrow \infty}a_{n} = L.
\end{equation*}
In more informal terms, if the sequence \(a_{n}\) is always between two sequences \(b_{n}\) and \(c_{n}\) for all \(n\text{,}\) and the limit is the same for both \(b_{n}\) and \(c_{n}\text{,}\) then the limit of \(a_{n}\) is the same.
