Sigma Notation

Learn and use sigma notation for series summation. Grade 11 Math 20-1 Alberta.

Lesson 3.6 of Sequences and Series in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Reading Sigma Notation. Learn the parts of sigma notation — the upper limit, lower limit, index, and general term — then expand a sigma expression and count its terms using a simple formula.
• Converting a Series to Sigma Notation. Go the other direction: given an arithmetic or geometric series in expanded form, identify the general term and limits, then write the entire series compactly in sigma notation.
• Sums and Working Backwards. Evaluate sigma expressions using arithmetic and geometric sum formulas — including infinite sums — and solve for unknown values hidden inside the limits of a sigma expression.

Inside the lesson — a free preview

What is Sigma Notation?

The Big Idea. The Greek letter (sigma) means "sum." Sigma notation is a compact shorthand for writing a series by specifying three things:

1. The general term — a formula for each term in the sequence.

2. The lower limit — the starting value of the index.

3. The upper limit — the ending value of the index.

For example, the series 3 + 9 + 27 + 81 + 243 + 729 is written in sigma notation as shown below.

How to Read It. Read _n=1^(6) 3^n as: "The sum of 3^n as n goes from 1 to 6."

To expand it, plug in n = 1, 2, 3, 4, 5, 6 into the general term and add:

3^1 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 = 3 + 9 + 27 + 81 + 243 + 729

The Parts of Sigma Notation

Note on the Index Letter. The choice of letter for the index does not matter — n, k, and i are all commonly used. What matters is that the same letter appears both below (as the index) and inside the general term.

Counting Terms

Number of Terms Formula. Number of terms = (upper limit) - (lower limit) + 1

Why the +1? If the index runs from 1 to 6, there are 6 terms (1, 2, 3, 4, 5, 6). Subtracting alone gives 6 - 1 = 5, so we must add 1 to count both endpoints.

Quick Examples. _n=1^(10): 10 - 1 + 1 = 10 terms

_k=4^(7): 7 - 4 + 1 = 4 terms

_n=5^(20): 20 - 5 + 1 = 16 terms

Worked example: Expand and Find the Sum

Example 1 — Expand _n=1^(4) 2^(n+2) and find the sum. Step 1 — Plug in each value of n:

n = 1: 2^(1+2) = 2^3 = 8

n = 2: 2^(2+2) = 2^4 = 16

n = 3: 2^(3+2) = 2^5 = 32

n = 4: 2^(4+2) = 2^6 = 64

Expanded form: 2^3 + 2^4 + 2^5 + 2^6

Step 2 — Add the terms:

8 + 16 + 32 + 64

Answer: Sum = 120

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