Summation Notation – Definition, Rules, And Examples

Summation notation provides a concise and compact representation of a long sum of the number that follows a specific pattern. It is denoted by the uppercase Greek letter Sigma (∑), derived from the Latin word “summa” It indicates the sum of terms in a sequence or series. It plays a crucial role in calculus to represent finite and infinite series.

In this article, we will define the definition of the summation notation in depth. We will learn how to write the sequences of numbers in summation notation. We will also learn how to expand the summation notation. In the example section, we will solve some examples of summation notation.

Definition of Summation Notation or Sigma Notation

Summation / sigma notation, is the easiest and most efficient method to write an extended sum of sequence elements. It simplifies the representation of large sums by using the sigma symbol (∑). It is widely used in mathematics for expressing the addition of terms that follow a specific pattern.

The sum a₁ + a₂+ a₃ + … + a_n is written in summation notation as:

Here sigma notation (∑) represent that we are summing or adding a series of terms together. The index variable (i) is a symbol that represents a number that changes as the summation is evaluated. The index variable “i” begin at 1 and increases until it reaches n. The summand (a_i) represents the expression or function that defines each term in the series.

Steps for writing the summation notation

Follow these steps to write a summation notion:

Determine the pattern or sequence of terms you want to sum.
Specify the indexed variable. It describes the position or order of each term in the sequence. It is commonly lowercase letters, like j, k, or i.
Evaluate the lower and upper limits of the summation. The lower limit is the starting point of the index, while the upper limit represents the ending point.
Write the sigma symbol (∑).
Write the lower limit down the sigma sign and the upper limit above the sigma symbol.
Write the expression or formula that defines the terms to be added together. This expression involves the indexed variable and may include other constants or variables.

Steps for expanding the summation notation

Summation notation expansion is the inverse process of summation notation writing. You can write the expanded form of summation notation by following the steps:

Start with the given summation notation
Substitute the first value of the index variable into the expression and write down the resulting term.
Repeat the previous step until you reach the final value of the index variable.
Combine all the written terms using the plus symbol between them.

Application of summation notation

Summation notation finds application in various fields of mathematics and statistics. Some applications of summation notation are given below:

Calculus and Integration: Summation notation is used in calculus to represent Riemann sums, which approximate the area under curves. It also plays a role in expressing infinite series used in calculus, such as power series.

Probability and statistics: Summation notation is used to calculate the expected values, probabilities, and sums of random variables.

Sequences and series: Summation is important for defining and studying arithmetic, geometric, and other types of sequences and series. It allows mathematicians to express and analyze patterns concisely.

Key points

If a_i and b_i are the functions, and k is any constant then,

⇒ ∑ _i=mⁿ (a_i ± b_i) = ∑ _i=mⁿ a_i ± ∑ _i=mⁿ b_i

⇒ ∑ _i=mⁿ (a_i × b_i) ≠ ∑ _i=mⁿ a_i × ∑ _i=mⁿ b_i

⇒ ∑ _i=mⁿ (a_i / b_i) ≠ ∑ _i=mⁿ a_i / ∑ _i=mⁿ b_i

⇒ ∑ _i=mⁿ (k × a_i) = k ∑ _i=mⁿ (a_i)

⇒ ∑ _i=mⁿ (a_i) = a_m + ∑ _i=m+1ⁿ (a_i)

Solved Examples of Summation Notation

Let’s solve some examples to understand it better.

Example 1.

Express the following Sum

1 + 4 + 9 + 16 + 25 + 36+ 49

In summation notation.

Solution.

Step 1: Observe the pattern. Notice that the terms in the sum are perfect squares of consecutive positive integers. I.e. 1² = 1, 2² = 4, 3² = 9… 7² = 49

Step 2: Let’s use “x” as the indexed variable.

Step 3: Since we want to start with the first term of the sum, the lower limit in this example is 1.

Step 4: We have seven terms in the given sum, so the upper limit is 7.

Step 5: Write the sigma symbol. Place the indexed variable “x” below the sigma symbol with a lower limit and the upper limit above the sigma symbol. write the term (x²) for the terms being summand.

Thus, the summation notation of the given sum is ∑ _x=1⁷ (x²)

Example 2.

Expand and simplify the following summation notation.

∑ _x=1⁴ (3x²)

Solution.

Here,

Upper limit = 4

Lower limit = 1

Substitute index values in the given express (3x²)

3(1)², 3(2)², 3(3)², 3(4)²

Place a positive sign between them, and we have,

∑ _x=1 (3x²) = 3(1)² + 3(2)² + 3(3)² + 3(4)²

= 3 [(1)² + (2)² + (3)² + (4)²]

It is the expanded form of the given summation notation. Now simplify the expression:

= 3 [1 + 4 + 9 + 16]

= 3 (30) = 90

∑ _x=1⁵ (3x²) = 90

A sigma calculator can be used as an alternative to solve the problems of summation notation instead of solving them manually to save time. Here is the above example solved by this calculator.

Conclusion

In this article, we have discussed the definition of summation notation. We learned how to write a summation of an extended sum and vice versa. We discussed the application of summation notation in mathematics and statistics. We covered rules for the Sigma notion.

Summation Notation – Definition, Rules, and Examples