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The summation operator is denoted by $\Sigma$. The summation operator is a mathematical notation used to represent the sum of numbers or terms. The summation is the total of all the terms added according to the specified range of values for the index.
Suppose, we have information about the height of students, such as 54, 55, 58, 60, 61, 45, 53.
Using variable and value notation one can denote the height of the students like
- First height in the information $X_1$, that is $X_1=54$
- Second height in the information $X_2$, that is $X_2=55$
- Last or nth information $X_n$, that is $X_n=53$.
In general, the variable and its values can be denoted by $X_i$, where $i=1,2,3, \cdots, n$.
The sum of all numeric information (values of the variable $X_1, X_2, \cdots, X_n$) can be totaled by $X_1+X_2+\cdots+X_n$. The short and useful summation for the set of values is $\sum\limits_{i=1}^n X_i$, where the symbol $\Sigma$ is a Greek letter and denotes the sum of all values ranging from $i=1$ (start) to $n$ (last) value.
The number written on top of $\Sigma$ is called the upper limit (Upper Bound) of the sum, below $\Sigma$, there are two additional components: the index and the lower bound (lower limit). On the right of $\Sigma$, there is the sum term for all the indexes.
Summation Operator
Consider the following example for the use of summing values using the Summation operator.
\begin{align*}
X_1 + X_2 + X_3 + \cdots X_n &= \sum\limits_{i=1}^{n} X_i\\
X_1Y_1 + X_2Y_2 + X_3Y_3 + \cdots X_nY_n &= \sum\limits_{i=1}^{n} X_iY_i\\
X_1^2 + X_2^2 + \cdots + X_3^2 + \cdots X_n^2 &= \sum\limits_{i=1}^n X_i^2\\
(X_1 + X_2 + X_3 + \cdots X_n)^2 &= \left( \sum\limits_{i=1}^{n} X_i \right)^2
\end{align*}
The following examples make use of the summation operator, when a number (constant) and values of the variable are involved.
\begin{align}
a+a+a+ \cdots + a = na&=\sum\limits_{i=1}^{n}a\\
aX_1 + aX_2 + aX_3 \cdots + aX_n &= a \sum\limits_{i=1}^n X_i\\
(X_1-a)+(X_2-a)+\cdots + (X_n-a) &= \sum\limits_{i=1}^n (X_i-a)\\
(X_1-a)^2+(X_2-a)^2+\cdots + (X_n-a)^2 &= \sum\limits_{i=1}^n (X_i-a)^2\\
[(X_1-a)+(X_2-a)+\cdots + (X_n-a)]^2 &= \left[\sum\limits_{i=1}^n (X_i-a)\right]^2
\end{align}
Properties of Summation Operator
The summation operator is denoted by the $\Sigma$ symbol. It is a mathematical notation used to represent the sum of a collection of (data) values. The following useful properties for the manipulation of the sum operator are:
1) Multiplying a sum by a constant
$$c\sum\limits_{i=1}^n x_i = \sum\limits_{i=1}^n cx_i$$
2) Linearity: The summation operator is linear meaning that it satisfies the following properties for constant $a$ and $b$, and sequence $x_n$ and $y_n$.
$$\sum\limits_{i=1}^N(ax_i + by_i) = a \sum_{i=1}^N x_n + b\sum\limits_{i=1}^N y_i$$
3) Splitting a sum into two sums
$$\sum\limits_{i=a}^n x_i = \sum\limits_{i=a}^{c}x_i + \sum_{i=c+1}^n x_i$$
4) Combining Summations: Multiple summations can be combined into a single summation:
$$\sum\limits_{i=1}^b x_n + \sum\limits_{i=b+1}^c x_i = \sum\limits_{i=1}^c x_i$$
5) Changing the order of individual sums in multiple sum expressions
$$\sum\limits_{i=1}^{m} \sum\limits_{j=1}^{n} a_{ij} = \sum\limits_{j=1}^{n}\sum\limits_{i=1}^{m} a_{ij}$$
6) Distributivity over Scalar Multiplication: The summation operator distributes over scalar multiplication
$$c\sum\limits_{i=1}^b x_i = \sum_{i=1}^b (cx_i)$$
7) Adding or Subtracting Sums
$$\sum\limits_{i=1}^a x_i \pm \sum_{i=1}^a y_i = \sum\limits_{i=1}^a (x_i \pm y_i)$$
8) Multiplying the Sums:
$$\sum\limits_{i_1=a_1}^{n_1} x_{i_1} \times \cdots \times \sum\limits_{i_n=a_n}^{n_n} x_{i_n} = \sum\limits_{i_1=a_1}^{n_1} \times \cdots \times \sum\limits_{i_1=a_1}^{n_n}x_{i_1}\times \cdots \times x_{i_n}$$
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