Sigma Notation: What is it and how to calculate it?

Sigma notation (A.K.A summation notation) is frequently used in mathematics to represent complex and lengthy series concisely. It expresses the sum of a sequence of terms defined by a function, with clear signs of the starting and ending values. In this article, we’ll explore the concept of sigma notation along with its properties and calculation.

What is Sigma Notation?

Sigma notation is a way of writing down sums. It is compact and a lot easier to write than the long form of a sum. Here is a breakdown of the Sigma notation:

_i=a∑^b f(i)

In this expression:

• The Σ (capital sigma) is the Greek letter sigma, which stands for 'sum'.
• The “i=a” under the sigma sign tells you where to start the sum. In this case, the index“i” starts at value ‘a’.
• The ‘b’ above the sigma sign tells you where to end the sum. The index i goes up to b.
• The f(i) to the right of the sigma sign is a function that gives you the term to be added. It could be a simple expression or a more complex formula.

Here is a concrete example:

_i=1∑⁴ (2i + 1)

In this case, we start with i=1 and end with i=4. The function we use to generate the terms is 2i+1. So the sum is:

(2×1 + 1) + (2×2 + 1) + (2×3 + 1) + (2×4 + 1) = 3 + 5 + 7 + 9 = 24

Sigma notation can be used for finite sums, like the example above, or for infinite series. For an infinite series, the upper bound b would be infinity (∞).

Properties of sigma notation:

There are several rules and properties of sigma notation that can be used to simplify sums and manipulate expressions. Here are some of the most common ones:

Constant Multiplication:

If you have a constant c multiplied by a function inside the sigma notation, you can take the constant outside of the sum:

_i=a∑^b c * f(i) = c * _i=a∑^b f(i)

Sum of Sums:

If you have a sum of two functions inside the sigma notation, you can split it into two separate sums:

_i=a∑^b [f(i) + g(i)] = _i=a∑^b f(i) + _i=a∑^b g(i)

Difference of Sums:

Similar to the sum of sums, if you have a difference of two functions inside the sigma notation, you can split it into two separate sums:

_i=a∑^b [f(i) - g(i)] = _i=a∑^b f(i) - _i=a∑^b g(i)

Changing the Index:

You can change the index of summation as long as you adjust the function and limits accordingly. For example:

_i=a∑^b [f(i)] = _j=0∑^n-1 f(j + 1)

Summation of a Constant:

If you are summing a constant value c, the result is c multiplied by the number of terms:

_i=a∑^b c = _i=a∑^b c(b – a + 1)

Sum of Consecutive Integers:

The sum of the first n positive integers is given by:

_i=a∑^b i = _i=a∑^b n(n + 1)/2

Sum of Consecutive Squares:

The sum of the squares of the first n positive integers is given by:

_i=a∑^b i² = _i=a∑^b n(n + 1).(2n + 1)/6

Telescoping Sum:

If you have a sum where each term cancels out with the next term, except for the first and last terms, then the sum simplifies to the first term minus the last term:

_i=a∑^b [f(i) - f(i + 1)] = f(a) – f(b + 1)

How to calculate sigma notation?

To calculate a sum in sigma notation, follow these steps:

Identify the Index of Summation and the Limits: Look at the variable used as the index of summation and identify the lower and upper limits of summation. The lower limit is the starting value of the index, and the upper limit is the ending value.

Write Out the Terms: Substitute each value of the index from the lower limit to the upper limit into the function given in the sigma notation, and write out each term.

Add the Terms: Finally, add up all the terms to get the sum.

Alternatively, you can search for an online sigma notation calculator that will solve all the complex sums for you.

Example 1:

Here's an example to illustrate the process:

_i=1∑⁴ [2i + 1]

The index of summation is i, with a lower limit of 1 and an upper limit of 4.

Substituting the values of i, we get the terms: 2⋅1+1=3, 2⋅2+1=5, 2⋅3+1=7, 2⋅4+1=9.

Adding up the terms, we get: 3+5+7+9=24. So, the sum of the series is 24.

Example 2:

If you have a series and you want to write it in sigma notation, you'll need to identify a pattern or formula that generates the terms of the series. Here's an example:

Consider the series: 3+6+9+12+15

We can observe that each term is a multiple of 3, and the terms are increasing by 3 each time. This pattern suggests that the i^th term of the series can be represented as 3i, where i is the index of summation.

The first term corresponds to i=1 and the last term corresponds to i=5. So, we can write the series in sigma notation as:

Applications of sigma notation:

Sigma notation is a very powerful and useful tool in various fields such as mathematics, physics, engineering, and even computer science. Here are some of the applications of sigma notation:

Mathematics:

• Sigma notation is used to represent sequences and series, including arithmetic and geometric series.
• It is also used in statistics to represent sums of data values, such as the sum of squared deviations.

Physics:

• In physics, sigma notation is often used to represent sums of forces, energies, or other physical quantities in a system.
• It can also be used to represent the sum of a series of waves or oscillations.

Engineering:

• In engineering, sigma notation is used to represent sums of loads, stresses, or other forces in a structure.
• It is also used in control engineering to represent the sum of errors in a control system.

Computer Science:

• In computer science, sigma notation is often used in algorithms and pseudocode to represent the sum of a series of values or the sum of a sequence of operations.
• It is also used in big-O notation to represent the sum of a series of terms in an algorithm's runtime or space complexity.

Conclusion

Overall, sigma notation is a very versatile and powerful tool that can be used to represent and simplify complex sums in a wide range of fields.