Summation Notation: Explore Its Basics in Mathematics

Summation comes from the Latin word “summa”, which means “sum”. Summation notation is a method of expressing the sum of a sequence of numbers more concisely and effectively. This notation is also called sigma notation. The Greek letter Sigma (Σ ) represents it. It plays a crucial role in calculus to represent the finite and infinite series.

We are going to explore the whole concept in this article. We will learn how to represent the sum of a sequence of numbers in summation notation. We will practice some examples to understand this concept in a better manner.

What is the summation notation?

Summation notation is the convenient and efficient method of writing sums of terms that follow a particular pattern. It involves the uppercase Greek letter sigma (Σ) to represent the sum of the sequence. It is used in different branches of mathematics, including calculus, linear algebra, and statistics

The sum of x₁ + x₂ + x₃ + … + x_n can be written in summation notation as:

Here,

• Σ expresses the sum of the terms of the sequence.
• x_i is the summand that refers to the expression or function defining each sequence term.
• i is the index variable that is used to indicate the position or order of the terms being summed.
• 1 is the lower limit and it is starting point of summation, while n is the upper limit and it is the ending value of summation.

Methods for writing Summation Notation:

You can follow these steps to write the sum of a sequence of numbers in summation notation:

1. Analyze the pattern of the given sequence.
2. Assign an index variable to represent the position of each term in the series.
3. Determine the lower and upper limits of summation notation.
4. Write the sigma symbol, with its lower and upper limits as subscripts and superscripts, respectively.

How to expand the summation notation?

The expanded form of sigma notation is the inverse method of writing sigma notation. To expand the summation notion, follow these steps:

1. Substitute each value of the index variable (beginning at the lower limit and increasing it until you reach the upper limit) into the general form.
2. Write a positive sign between all obtained terms.
3. Simplify the expression if it is necessary.

Application of summation notation in mathematics

Some common applications of sigma notation in mathematics are here:

• It is used to express general sequences and series. The sum of numbers in a sequence can be written as ∑_{n = 1} ^∞ (n).
• In calculus, it is used to represent Riemann sums, which approximate the area under a curve. The sum of a large number of small intervals is written as ∑_{i = 1} ⁿ f(x_i) Δx.
• In probability and statistics, it is used to calculate the sum of probabilities or statistical measures. The expected value of a discrete random variable can be written, as ∑ (x) *P (x). Where x represents the possible outcomes and P (x) is the corresponding probability.

Properties of Sigma Notation

If x_i and y_i are two functions, k is an arbitrary constant. Then the following summation properties hold,

1. Commutative properties hold in summation notation with respect to addition and subtraction

∑_i=mⁿ (x_i ± y_i) = ∑_i=mⁿ x_i ± ∑_i=mⁿ y_i

2. Product and Quotient properties do not hold in summation notation

∑_i=mⁿ (x_i × y_i) ≠ ∑ _i=mⁿ x_i × ∑_i=mⁿ y_i

∑_i=mⁿ (x_i / y_i) ≠ ∑ _i=mⁿ x_i /  ∑_i=mⁿ y_i

3. Summation notation for Constant

∑ _i=mⁿ (k × x_i) = k ∑ _i=mⁿ (x_i)

4. Sigma notation break

∑ _i=mⁿ (x_i) = x_m + ∑ _i=m+1ⁿ (x_i)

Solved Examples

Here are some solved examples of summation notation:

Example 1.

Express the following sum of sequence in summation notation.

9 + 16 + 25 + 36 + 49,… + 144

Solution.

Step 1: Find out the pattern of the given sequence. We can observe that the terms in the sequence are perfect squares. I.e. 3² + 4² + 5² +… + 12²

Step 2: Choose “i” as the index variable.

Step 3: First-term 3 denotes the lower limit of summation and the last value 12 indicates the upper limit.

Step 4: Use the sigma symbol. Write an indexed variable “i” below the sigma symbol with a lower limit. Place the upper limit above the sigma symbol. The term (i²) will be used as a summand.

Thus, the summation notation of the given sequence is ∑ _x=3¹² (i²)

Example 2.

Write the Expanded form of the following summation notation. Also simply.

_{∑ x=2}⁶ (4x³)

Solution.

Given,

Upper limit = n = 6

Lower limit = i = 2

By using summation properties of summation, we can write

∑ _x=2⁶ (4x³) = 4 ∑ _x=2⁶ (x³)

Put index values in the given expression (4x³), and write a positive symbol between them, and we get,

∑_x=2⁶ (4x³) = 4 ∑ _x=2⁶ (x³) = 4 [2³ + 3³ + 4³ + 5³ + 6³]

∑_x=2⁶ (4x³) = 4 [8 + 27 + 64 + 125 + 216]

That is the expanded form of the given sigma notation. Now simplify it:

∑ _x=2⁶ (4x³) = 4 (440) = 1760

Thus, ∑_x=2⁶ (4x³) = 1760

A sigma calculator can be used to expand the summation notation to find the sum of the given function with steps to get rid of time-consuming calculations.