The Sum of series or Summation of sequences can be defined as adding up the set of values/numbers in an ordered form or series. Summation of sequences has two types of sequences known as finite and infinite set of sequences.

The finite sequence will have an upper limit and lower limit (first and last values) and the infinite sequences will infinitely continue in the series. The meaning of Summation (Σ) is just to “add up.” For example, let’s say that you had 4 items in a data set: 1,2,5,7 you can think that these values are placed on the x-axis also called x-values. If you were asked to add up all of the items in summation notation, you would see that the:

$$\text{Summation}\;\sum(x)\;\text{which is equals to}\;1\;+\;2\;+\;5\;+\;7\;=\; 15.$$

When we are using summation notation,X1 means “the first x-value”, X2 means “the second x-value” and so on till the end. For example, let’s say that you had a list of weights: 50kg, 100kg, 150kg and 200kg. The weights and their corresponding x-values are:

X1: 50kg

X2: 100kg

X3: 150kg

X4: 200kg

Here is the simple form of Summation Σ(x):

$$\text{Summation}\;\sum(x)=\;\sum_{i=1}^n X_i=X_1+X_2+X_3+…+X_n$$

The “i=1” at the base of Σ means that it is the start of your first x-value. This would be X1 (50kg as shown in the above example). The “n” at the top of Σ means that the end number of the given values”. In statistics, n is the number of items in the given data set. So, what this summation is asking you to do is “sum up all of your x-values from the first(“i=1”) to the last (n).” For this set of data, that would be:

$$50\;kg\;+\;100\;kg\;+\;150\;kg\;+\;200\;kg\;=\;500\;kg$$

Note: If you see a number above this sign Σ, instead of n, it means that you have to sum up the values to that certain number. For example, a “3” above the Σ means to add up the third item (X3) in the set but you don’t need to worry about calculation. Calculatored introduce Summation Calculator for calculation of Summation notation.

## Sigma Notation

The Sigma notation is appearing as the symbol S, which is derived from the Greek upper-case letter, S. The sigma symbol (S) indicate us to sum the values of a sequence. A typical value of the sequence which is going to be add up appears to the right of the sigma symbol.

The variable of sigma notation is the variable which is going to add up.

The variable of sigma notation is represented by an index which is placed below the sigma symbol. The index is typically represented by i. (j and t are the other common possibilities for representation of the index in sigma notation). The index appears in the form of expression i = 1. The index assumes that values of the sequence starting with the value on the right-hand side of the equation and ending with the value above the sigma notation.

- The starting point for the summation notation is known as lower limit of summation notation.
- The stopping point for the summation notation is known as upper limit of summation notation.

Here is the more explanation of sigma symbol.

## Sigma Calculator

To calculate Sigma notation or summation notation we must have these things:

- Sigma symbol
- Upper limits
- Lower limits
- N number (what to sum)
- Sequence of value

Example:

$$\text{Summation}\;\sum(x)=\;\sum_{i=1}^n X_i=X_1+X_2+X_3+…+X_n$$

We have 4 values:

- X
_{₁}=10 - X
_{₂}=20 - X
_{₃}=30 - X
_{₄}=40

BY using the above equation, we can calculate the summation notation.

$$\text{Summation}\;\sum(x)=\;\sum_{i=1}^n X_i=10+20+30+40=100$$

After adding up all the values we got summation notation.

## Summation Calculator

Calculatored introduce Summation Calculator for calculation of Summation notation:

Where,

You have to enter three things:

- Upper limit
- Function
- Lower limit

After you entered the values then press “CALCULATE” button and Summation Calculator will calculate the summation of the sequence.

**Here is an example to calculate the summation of the function through calculatored’s Summation Calculator.**

After entered the corresponding values:

- Upper limit =5
- Function = 4x
- Lower limit=1

Press “CALCULATE” button and Summation Calculator will calculate the summation of the sequence on the right side of the Summation Calculator in the status block.