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

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.

\( \begin{align} \displaystyle
S_{n} &= (2 \times 1)+(2 \times 2)+(2 \times 3)+\cdots+(2 \times n) \\
&= \sum_{k=1}^{n}{2k} \\
\end{align} \)

Example 2

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

\( \begin{align} \displaystyle
\sum_{k=1}^{6}{(k+4)} &= (1+4)+(2+4)+(3+4)+(4+4)+(5+4)+(6+4) \\
&= 5+6+7+8+9+10 \\
&= 45 \\
\end{align} \)

Example 3

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

\( \begin{align} \displaystyle
&= 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 \\
&= \sum_{k=1}^{6}{2^k} \\
\end{align} \)

Example 4

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

\( \begin{align} \displaystyle
\sum_{k=3}^{100}{2} &= \overbrace{2+2+2+\cdots+2}^{98} \\
&= 2 \times 98 \\
&= 196 \\
\end{align} \)

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

\( \begin{align} \displaystyle
\text{LHS} &= \sum_{k=1}^{n}{(a_{k} + b_{k})} \\
&= (a_{1}+b_{1})+(a_{2}+b_{2})+(a_{3}+b_{3})+\cdots+(a_{n}+b_{n}) \\
&= (a_{1}+a_{2}+a_{3}+\cdots+a_{n}) + (b_{1}+b_{2}+b_{3}+\cdots+b_{n}) \\
&= \sum_{k=1}^{n}{a_{k}} + \sum_{k=1}^{n}{b_{k}} \\
&= \text{RHS}
\end{align} \)

Example 6

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

\( \begin{align} \displaystyle
\text{LHS} &= \sum_{k=1}^{n}{Au_{k}} \\
&= Au_{1}+Au_{2}+Au_{3}+\cdots+Au_{n} \\
&= A(u_{1}+u_{2}+u_{3}+\cdots+u_{n}) \\
&= A\sum_{k=1}^{n}{u_{k}} \\
&= \text{RHS} \\
\end{align} \)

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