A sequence is an ordered list of terms.
These are commonly maps from natural numbers to real (or complex) numbers.
We can use \(a_i=f(i)\) to denote this.
If \(f(i)\) is defined on all \(i\in \mathbb{N}\) then the sequence is infinite. Otherwise it is finite.
If a sequence is defined on \(n\in \mathbb{N}\) and \(n\ne 0\) then the sequence must be defined on \(n-1\).
For example \(a_0, a_1, a_2,...\) is a sequence, but \(a_1,a_2,...\) is not.
A monotone sequence is one where each element is succeeded ordinally by the next entry.
For example:
\(<1,2,3,6,7>\) is monotone
\(<1,2,3,3,4>\) is not monotone
An increasing sequence is one where:
\(\forall m \in \mathbb{N} \forall n\in \mathbb{N} [m > n \leftrightarrow a_m \ge a_n]\)
A strictly increasing sequence is one where:
\(\forall m \in \mathbb{N} \forall n\in \mathbb{N} [m > n \leftrightarrow a_m > a_n]\)
A decreasing sequence is one where:
\(\forall m \in \mathbb{N} \forall n\in \mathbb{N} [m > n \leftrightarrow a_m \le a_n]\)
A strictly decreasing sequence is one where:
\(\forall m \in \mathbb{N} \forall n\in \mathbb{N} [m > n \leftrightarrow a_m < a_n]\)
All strictly decreasing sequences are decreasing, and all strictly increasing sequences are increasing.
A monotone sequence is one which is either increasing or decreasing.
A subsequence of a sequence is the original sequence with some elements of the original removed, not changing the order.
For example:
\(<1,3,5>\) is a subsequence of \(<2,1,3,4,7,5>\)
A series is the summation of a sequence. For a series \(a_n\) there is a corresponding series:
\(s_n=\sum_{i=0}^na_n\)
Where:
\(\sum_{i=0}^na_i=a_0+a_1+a_2+...+a_n\)
If all members of a sequence are multiplied by a constant, so is each member of the series.
We can take constants out of the series:
\(s_n=\sum_{i=0}^na_i\)
\(s_n=\sum_{i=0}^ncb_i\)
\(s_n=a\sum_{i=0}^nb_i\)
If all elements of a sequence are the same, then the series is a multiple of that constant.
\(s_n=\sum_{i=0}^na_i\)
\(s_n=\sum_{i=0}^nc\)
\(s_n=nc\)
Consider a sequence \(a_i=b_i+c_i\).
\(s_n=\sum_{i=0}^na_i\)
\(s_n=\sum_{i=0}^n(b_i+c_i)\)
We can then split this out.
\(s_n=\sum_{i=0}^nb_i+\sum_{i=0}^nc_i\)
\(\sum_{i=0}^na_i=a_0+\sum_{i=1}^na_i\)
\(\sum^n_{i=0} \sum^m_{j=0} a_i=n\sum^m_{j=0} a_i\)
\(\sum^n_{i=0} \sum^m_{j=0} a_i b_j=\sum^n_{i=0}a_i\sum^m_{j=0}b_i\)
A product is a repeated multiplication of a sequence.
\(p_n=\prod_{i=0}^ns_i\)
We can take constants out of the product.
\(p_n=\prod_{i=0}^nca_i\)
\(p_n=a^{n}\sum_{i=0}^na_i\)
If \(a_i=c\) then the summation is then of the form:
\(p_n=\prod_{i=0}^nc\)
\(p_n=c^{n}\prod_{i=j}^n1\)
\(p_n=c^{n}\)
If a sequence is the product of to other sequences then the product of the sequence is equal to the product of the two individual sequences.
\(p_n=\prod_{i=0}^na_i\)
\(p_n=\prod_{i=0}^nb_ic_i\)
\(p_n=\prod_{i=0}^nb_i\prod_{i=0}^nc_i\)
A factorial is a a product across natural numbers. That is:
\(n!:=\prod^n_{i=0}i\)
Let’s prove that:
\(\sum_{i=0}^n i= \dfrac{n(n+1)}{2}\)
We use the inference rules Modus Ponens, which says that if \(X\) is true, and \(X\rightarrow Y\) is true, then \(Y\) is true.
We know this is true for \(n=0\):
\(0=\dfrac{0(0+1)}{2}\)
\(0=0\)
We can also prove that if it true for \(n\), it is true for \(n+1\).
\(\sum_{i=0}^{n+1} i= \dfrac{(n+1)(n+2)}{2}\)
\((n+1)+\sum_{i=0}^{n} i= \dfrac{n^2 +3n +2}{2}\)
If it is true for \(n\), then:
\((n+1)+\dfrac{n(n+1)}{2}= \dfrac{n^2 +3n +2}{2}\)
\(\dfrac{n^2+3n+2}{2}= \dfrac{n^2 +3n +2}{2}\)
\(1=1\)
So we know that it is true for \(n=0\), and if it is true for \(n\), then it is true for \(n+1\). As a result it is true for all natural numbers.
A function \(f(x)\) on set \(X\) is bounded if:
\(\exists M\in \mathbb{R} [\forall x\in X f(x)\le M]\)
A bounded sequence is a special case of a bounded function where:
\(X=\mathbb{N}\)
That is, a sequence is bounded by \(M\) iff:
\(\forall n\in \mathbb{R} |f(a_n)|\le M\)