Summation Notation & Sigma Function

Contents:

1. What is summation Notation?
2. Summation Notation examples
3. Sigma Function
4. Useful summation identities

See also: Einstein Summation.

What is summation notation?

Some formulas require the addition of many variables; summation notation is a shorthand way to write a concise expression for a sum of a variable’s values. The formula contains the uppercase Greek letter sigma (Σ), which is why summation notation is sometimes called sigma notation.
The “a_i” in the above sigma notation is saying that you sum all of the values of “a”. In other words, you’re adding up a series of values: a₁, a₂, a₃, …, a_x.

• i is the index of summation. It doesn’t have to be “i”: it could be any variable (j, k, x etc.).
• a_i is the ith term in the sum.
• n and 1 are the upper and lower bounds of summation. I’m using “1” here as an example: the lower bound could be an integer less than or equal to n.

The notation can be broken down into parts:

Symbol	Meaning
Greek capital letter sigma (Σ)	Sum (add) the terms
Variable below Σ (such as i, j, k, m, or n).	Index of the sum, representing the data points in the set (e.g., i = 1, i = 2, …_
Number below Σ	Lower limit of the sum (where to start adding)
Number above Σ	Upper limit of the sum (where to stop adding)
Number or variable to the right of Σ	Terms to be summed

Summation Notation examples

The following image shows sigma notation for adding up a series of digits from 1 to 6. The lower bound (1) and upper bound (6) are below and above the sigma, respectively. Basically, you start adding at 1 and stop when you get to 6:
In the following example, “k” is the index of summation because there’s a “k” in the formula.
It’s telling you to start at k = 1 (lower bound) and keep on summing. It might seem that you keep on adding infinitely, but you’ll usually stop when your function either converges (settles on a certain number) or clearly diverges (shows no hope of convergence).

The index of the sum initializes a variable set to a certain number. We continue adding, increasing j by one from that initial value [1]. Sometimes, there might not be a stated stopping point (upper bound). If this is the case, it may appear that you add infinitely, but you typically stop when your function either converges (reaches a certain number) or diverges (shows no indication of convergence).

Example 1: Find the sum  

Solution:

1. Find the starting point and stopping point.
   • The starting point is given at the bottom of sigma (Σ) as 1 and the stopping point is given at the top of sigma (Σ) as 5. The start and stop points tell us we are adding all terms from the first to the fifth.
2. Calculate the individual terms by replacing the k with the values 1, 2, 3, 4, 5 to find a₁, a₂, a₃, a₄, a₅.
   • a₁ = 1
   • a₂ = 2
   • a₃ = 3
   • a₄ = 4
   • a₅ = 5.
3. Add up the terms in Step 2 to get the sum: 1 + 2 + 3 + 4 + 5 = 15.

Example 2: Find the sum

Solution(note that this is a constant function with the value of 3):

1. Find the starting point and stopping point.
   • The starting point is given at the bottom of sigma (Σ) as 1 and the stopping point is given at the top of sigma (Σ) as 5.
2. Calculate the individual terms by replacing the 3 with the values 3, 3, 3, 3, 3 to find a₁, a₂, a₃, a₄, a₅.
   • a₁ = 3
   • a₂ = 3
   • a₃ = 3
   • a₄ = 3
   • a₅ = 3.
3. Add up the terms in Step 2 to get the sum: 3 + 3 + 3 + 3 + 3 = 15.

Example 3: Find the sum

Solution:

1. Find the starting point and stopping point.
   • The starting point is given as 3 and the stopping point is given as 7.
2. Calculate the individual terms by replacing the k with the values 3, 4, 5, 6, 7 to find a₃, a₄, a₅, a₆, a₇.
   • a₃ = (-1)³ (3) = -3
   • a₄ = (-1)⁴ (4) = 4
   • a₅ = (-1)⁵ (5) = -5
   • a₆ = (-1)⁶ (6) = 6
   • a₇ = (-1)⁷ (7) = -7
3. Add up the terms in Step 2 to get the sum: -3 + 4 + -5 + 6 + -7 = -5.

Using summation notation: rectangles example (calculus)

Sigma notation can be used to evaluate sums of rectangular areas. You can think of the bounds of summation here as where your rectangles start, and where they end. Example problem: Evaluate the sum of the rectangular areas in the figure below. Use sigma notation:
Step 1: Multiply the lengths of the base by the height of each rectangle.

• 1 * ¹⁄₃ = ¹⁄₃
• 1 * ¹⁄₄ = ¹⁄₄
• 1 * ¹⁄₅ = ¹⁄₅

Step 2: Add up the numbers you calculated in Step 1: ¹⁄₃ + ¹⁄₄ + ¹⁄₅ = ⁴⁷⁄₆₀. Step 3: Write the summand ¹⁄_k to the right of the sigma. The variables i, j, and k are usually used instead of x:
Step 4: Write the place where the summation ends at the top of Σ. This is a right-hand Riemann sum and so the measurement ends at the right of the last rectangle, at x = 5.

Step 5: Write the place where the summation starts at the bottom of Σ, after the index of summation (in this case, the index of summation is k).
That’s it!

Useful summation identities

1. Adjusting summation bounds. Use when the data points in a set are not evenly spaced. For example, if you have a set of even numbers, you can adjust the summation bounds so that you only sum over even numbers.
2. Finite geometric series — a geometric series is a sequence of numbers where each term is equal to the previous term multiplied by a constant value.
3. Factor out a constant. This can make the summation easier to calculate.
4. Gauss’s identity: expresses a sum in terms of a smaller number of sums.
5. Infinite geometric series (only valid when −1 < x < 1).
6. Sum of squares: expresses the sum of the squares of a set of numbers in terms of a smaller number of sums. For example, we can express the sum of the squares of all the numbers from 1 to 10 as ∑k10 (k2 + k) = (10 * 11 * 21) / 6 = 385.
7. Summation of a constant: explains what to do with a constant (as in example #2 earlier in this article).
8. Splitting a sum: splits sums into two or more parts for easier calculation.

What is a Sigma Function?

The “sigma function” may refer to:

1. The sum of positive divisors of some number n, or
2. The Weierstrass sigma function.

As a Sum of Positive Divisors

In number theory, the Sigma Function (denoted σ(n) or Σ(n)) of a positive integer is the sum of the positive divisors of n. For example, the number 3 has two positive divisors (1, 3) with a sum of 1 + 3 = 4. So: σ(3) = 4. A few more examples:

• σ(6) = (1 + 2 + 3) = 6,
• σ(12) = (1 + 2 + 3 + 4 + 6) = 16,
• σ(15) = (1 + 3 + 5) = 9.

Interestingly, the sigma function for any prime number is just that number plus one. That’s because primes are only divisible by itself and one.

The Weierstrass Sigma Function

The Weierstrass sigma function, usually denoted σ(z), is used in complex analysis and elliptic function theory (an elliptic function is a doubly periodic function). It was named after German mathematician Karl Weierstrass (October 1815 – 19 February 1897) and appeared in several of his works. The Weierstrass elliptic functions, which includes sigma, are in a relatively simple form and are also called p-functions, with a stylized letter P: ℘. The Weiestrass Sigma function is defined as:

1. σ(z) is an odd function of z. Odd functions are symmetrical about the origin.
2. σ(z) is an entire function.

References

1. CSE 373, Spring 2019: Summation Identities