27. Sigma Notation

Dated: 30-10-2024

The Greek letter \(\sum\) is called sigma or summation and is used to represent a lengthy sum into a more compact form.

Example

Consider a sequence as such

\[1^2 + 2^2 + 3^2 + 4^2 + \ldots\]

Notice how there is a gradual increase in the terms? we can write it as

\[\sum_{i = 1}^{\infty} i^2\]

The lower limit \(i = 1\) states that "\(i\) starts from \(1\)".
The upper limit \(\infty\) means "\(i\) goes till \(\infty\)".
The letter \(i\) is called the index variable.

Example

We can also change the lower limit by changing the index variable as such.

\[\sum_{k = 3}^{7} = 2^{k - 2}\]

We will change \(k = 3\) to \(j = k - 3\).
This ensures that as \(k\) starts from \(3\), \(j\) starts from \(0\) and as \(k\) stops at \(7\), \(j\) stops at \(4\).

\[\sum_{j = 0}^{4}2^{(j + 3) - 2}\]

\[\sum_{j = 0}^{4}2^{j + 1}\]

Example

This index variable works for everything.

\[\sum_{i = 1}^{\infty} i^2 = 1^2 + 2^2 + 3^2 + \ldots\]

\[\sum_{i = 1}^{\infty} 2^i = 2^1 + 2^2 + 2^3 + \ldots\]

\[\sum_{i = 1}^{\infty} a_i = a_1 + a_2 + a_3 + \ldots\]

Properties

• \[\sum_{i = 1}^{n} c \cdot a_i = c \cdot a_1 + c \cdot + a_2 + \ldots c \cdot a_n= c (a_1 + a_2 + \ldots + a_n) = c \sum_{i = 1}^{n}a_i\]
• \[\sum_{i=1}^{n}(a_{i}+b_{i})=\sum_{i=1}^{n}a_{i}+\sum_{i=1}^{n}b_{i}\]
• \[\sum_{i=1}^{n}(a_{i}-b_{i})=\sum_{i=1}^{n}a_{i}-\sum_{i=1}^{n}b_{i}\]

Theorems

• \[\sum_{i=1}^{n} i = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\]
• \[\sum_{i=1}^{n} i^2 = 1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\]
• \[\sum_{i=1}^{n} i^3 = 1^3 + 2^3 + 3^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2\]

Example

\[\sum_{i=1}^{30} i(i+1)\]

Using the distributive property

\[ = \sum_{i=1}^{30}(i^{2}+i)\]

Using the properties of sigma

\[ = \sum_{i=1}^{30}i^{2}+\sum_{i=1}^{30}i\]

using the theorems

\[= \frac{(30)(31)(61)}{6}+\frac{30(31)}{2}=9920\]

References

Read more about notations and symbols.