This set of notes explains the broadly accepted shorthand notation for sums and products of sequences.
Summation Notation
It is convenient to use a shorthand notation for sums of sequences.
The capital Greek letter \(\Sigma\) “sigma” is used to denote a sum or summation. Its use in mathematical notation for a sequence summation is defined as follows:
If \(m\) and \(n\) are integers and \(m \leq n\), the notation;
$$\sum_{k=m}^{n} a_{k}$$
is the sum of all terms of \(a\) as shown here in the expanded form:
$$a_m + a_{m+1} + a_{m+2} + \cdots + a_n$$
Parameters:
- \(k\): Index of the summation term.
- \(m\): Lower limit of the summation terms.
- \(n\): Upper limit of the summation terms.
Product Notation
Likewise there is a shorthand notation for products of sequences.
The capital Greek letter \(\Pi\) “pi” is used to denote a product. Its use in mathematical notation for the product of a sequence is defined as follows:
$$\prod_{i=m}^{n} a_i$$
is the product of all terms of \(a\) as shown here in the expanded form:
$$a_m * a_{m+1} * a_{m+2} * \cdots * a_n$$
Parameters:
- \(i\): Index of the product term.
- \(m\): Lower limit of the product terms.
- \(n\): Upper limit of the product terms.