Sigma Notation

Another mathematical device that is widely used in sequences and series is called $\textit{sigma notation}$. The Greek letter, $\sum$ (capital sigma), is used to indicate the sum of a sequence.

For example: $$\sum_{n=1}^{10}{n^2} = 1^2 + 2^2 + 3^2 + \cdots + 10^2$$ The limits of the sum, the numbers on the bottom and top of the $\sum$, indicate the terms that are to be included in the sum. When there is no chance of misinterpretation, the lower limit, $n=1$, may be abbreviated to $1$.

Sigma Notation

Sigma Notation

A $\textit{series}$ is the sum of the terms of a sequence.
For the $\textit{finite}$ sequence $\{u_{n}\}$ with $n$ terms, the corresponding series is $u_{1}+u_{2}+u_{3}+\cdots+u_{n}$.
The sum of this series is $S_{n}=u_{1}+u_{2}+u_{3}+\cdots+u_{n}$ and this will always be a finite real number.
$$\require{color} \color{red} S_{n}=\sum_{k=1}^{n}{u_{k}} = u_{1} + u_{2} + u_{3} + \cdots + u_{n}$$ For the $\textit{infinite}$ sequence $\{u_{n}\}$, the corresponding series is $u_{1}+u_{2}+u_{3}+\cdots+u_{n}+\cdots$.
In many cases, the sum of an infinite series cannot be calculated, In some cases, however, it does converge to a finite number. $$\require{color} \color{red} S_{\infty}=\sum_{k=1}^{\infty}{u_{k}} = u_{1} + u_{2} + u_{3} + \cdots + u_{n} + \cdots$$

Note 1

The lower limit does not have to be always $1$.
\( \begin{align} \displaystyle \sum_{k=4}^{7}{(2k+1)} &= (2\times 4+1) + (2\times 5+1) + (2\times 6+1) + (2\times 7+1) \\ &= 48 \\ \end{align} \)

Note 2

It is important to take the proper order of sequences of calculation such as braces and brackets. \( \begin{align} \displaystyle \sum_{k=1}^{4}{(2k+1)} &\ne \sum_{k=1}^{4}{2k}+1 \\ \text{LHS} &= \sum_{k=1}^{4}{(2k+1)} \\ &= (2 \times 1 + 1)+(2 \times 2 + 1)+(2 \times 3 + 1)+(2 \times 4 + 1) \\ &= 24 \\ \text{RHS} &= \sum_{k=1}^{4}{2k}+1 \\ &= (2 \times 1)+(2 \times 2)+(2 \times 3)+(2 \times 4)+1 \\ &= 21 \\ \therefore \sum_{k=1}^{4}{(2k+1)} &\ne \sum_{k=1}^{4}{2k}+1 \\ \end{align} \)

Example 1

Consider the sequence $2, 4, 6, \cdots$. Write down an expression for $S_{n}$, the sum of the first $n$ terms, using sigma notation.

Example 2

Expand and evaluate \(\displaystyle \sum_{k=1}^{6}{(k+4)} \).

Example 3

Write down an expression using sigma notation of the sequence $2+4+8+16+32+64$.

Example 4

Evaluate \( \displaystyle \sum_{k=3}^{100}{2}\).

Properties of Sigma Notation

$$\require{color} \color{red}\sum_{k=1}^{n}{(a_{k} + b_{k})} = \sum_{k=1}^{n}{a_{k}} + \sum_{k=1}^{n}{b_{k}}$$ $$\require{color} \color{red}\sum_{k=1}^{n}{Au_{k}} = A\sum_{k=1}^{n}{u_{k}}$$ where $\color{red}A$ is a constant.

Example 5

Prove \( \displaystyle \sum_{k=1}^{n}{(a_{k} + b_{k})} = \sum_{k=1}^{n}{a_{k}} + \sum_{k=1}^{n}{b_{k}} \).

Example 6

Prove \( \displaystyle \sum_{k=1}^{n}{Au_{k}} = A\sum_{k=1}^{n}{u_{k}} \).

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