This set of notes explains the broadly accepted shorthand notation for sums and products of sequences.

Summation Notation

It is often convenient to use this shorthand notation for the sum of a sequence.

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.

\(\LaTeX\) syntax:

$$\sum_{k=m}^{n} a_{k}$$

Product Notation

Here is a similar type of shorthand notation for the product of a sequence.

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.

\(\LaTeX\) syntax:

$$\prod_{i=m}^{n} a_i$$