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Sigma Notation — Definition, Formula & Examples

Sigma Notation
Continued Sum

A notation using the Greek letter sigma (Σ) that allows a long sum to be written compactly.

 

Example: Sum from k=5 to 100 of k² = 5² + 4² + 5² + 6² + … + 100²

 

 

See also

Series

Key Formula

∑i=mnai=am+am+1+am+2+⋯+an\sum_{i=m}^{n} a_i = a_m + a_{m+1} + a_{m+2} + \cdots + a_n
Where:
  • Σ\Sigma = The Greek capital letter sigma, meaning 'sum'
  • ii = The index variable (also called the counter or dummy variable)
  • mm = The lower limit — the starting value of the index
  • nn = The upper limit — the ending value of the index
  • aia_i = The general term — a formula that depends on the index i

Worked Example

Problem: Evaluate the sum: ∑ from i = 1 to 5 of 2i.
Step 1: Write out the notation and identify the parts. The index i starts at 1 and ends at 5, and the general term is 2i.
∑i=152i\sum_{i=1}^{5} 2i
Step 2: Substitute each value of i from 1 through 5 into the expression 2i to list every term.
2(1)+2(2)+2(3)+2(4)+2(5)2(1) + 2(2) + 2(3) + 2(4) + 2(5)
Step 3: Evaluate each term.
2+4+6+8+102 + 4 + 6 + 8 + 10
Step 4: Add all the terms together.
2+4+6+8+10=302 + 4 + 6 + 8 + 10 = 30
Answer: The sum equals 30.

Another Example

Problem: Write the sum 1 + 4 + 9 + 16 + 25 in sigma notation.
Step 1: Identify the pattern. The terms are 1², 2², 3², 4², 5², so the general term is i².
1=12,  4=22,  9=32,  16=42,  25=521 = 1^2,\; 4 = 2^2,\; 9 = 3^2,\; 16 = 4^2,\; 25 = 5^2
Step 2: Determine the lower and upper limits. The index i runs from 1 to 5.
Step 3: Write the sum in sigma notation.
∑i=15i2\sum_{i=1}^{5} i^2
Answer: The sum 1 + 4 + 9 + 16 + 25 can be written as ∑ from i = 1 to 5 of i².

Frequently Asked Questions

How do you read sigma notation out loud?
You read ∑i=152i\sum_{i=1}^{5} 2i as 'the sum from i equals 1 to 5 of 2i.' Some people say 'the summation of 2i as i goes from 1 to 5.' Both are standard.
Does the index variable have to be the letter i?
No. You can use any letter — j, k, n, or anything else. The index variable is a 'dummy variable,' meaning the final value of the sum does not depend on which letter you choose. For example, ∑i=13i2\sum_{i=1}^{3} i^2 and ∑k=13k2\sum_{k=1}^{3} k^2 both equal 14.

Sigma Notation (Summation) vs. Pi Notation (Product)

Sigma notation ∑\sum represents repeated addition of terms, while pi notation ∏\prod represents repeated multiplication of terms. For example, ∑i=14i=1+2+3+4=10\sum_{i=1}^{4} i = 1+2+3+4 = 10, whereas ∏i=14i=1×2×3×4=24\prod_{i=1}^{4} i = 1 \times 2 \times 3 \times 4 = 24. The structure — index, lower limit, upper limit, general term — is identical in both.

Why It Matters

Sigma notation is essential whenever you need to express large or even infinite sums without writing out every single term. It appears throughout algebra, statistics (for computing means and standard deviations), calculus (Riemann sums, Taylor series), and computer science (analyzing algorithm complexity). Mastering it lets you read and manipulate formulas far more efficiently than writing long expanded sums.

Common Mistakes

Mistake: Starting the index at the wrong value, typically assuming it always starts at 1.
Correction: Always check the lower limit. Sums can start at 0, 2, or any integer. For example, ∑i=03i\sum_{i=0}^{3} i includes the term where i = 0, giving 0 + 1 + 2 + 3 = 6, not 1 + 2 + 3 = 6.
Mistake: Forgetting to include the term at the upper limit.
Correction: The upper limit is included in the sum. In ∑i=14i\sum_{i=1}^{4} i, you must add the term when i = 4. The sum is 1 + 2 + 3 + 4 = 10, not 1 + 2 + 3 = 6.

Related Terms

  • Series — A series is the sum of a sequence's terms
  • Sum — The result of adding numbers together
  • Sequence — An ordered list of terms that sigma notation sums
  • Arithmetic Series — Common series type often written in sigma notation
  • Geometric Series — Another common series expressed with sigma notation
  • Greek Alphabet — Σ (sigma) is a letter from this alphabet
  • Index — The counter variable beneath the sigma symbol