Series and Sigma Notation
A series adds up the terms of a sequence.
About this topic
A series is the sum of the terms of a sequence. Sigma notation Σ is shorthand: the symbol with limits below and above tells you which term numbers to add. For example, Σ from k = 1 to 4 of 2k means 2 + 4 + 6 + 8 = 20.
There are formulas for the sum of an arithmetic series, Sₙ = n(a₁ + aₙ)/2, and a geometric series, Sₙ = a₁(1 − rⁿ)/(1 − r). The first formula is the one a young Gauss is said to have used to add 1 to 100 instantly: 100 · 101 / 2 = 5050. The idea is to pair the first and last terms, which always have the same sum.
Series are used to total payments, add up distances in a bouncing ball problem, calculate the total seats in a stadium, and in higher math, to build functions and approximate values. When the ratio of a geometric series is between −1 and 1, the infinite sum converges to a1/(1 − r).
Read the limits carefully: the number below the sigma is where you start, and the one above is where you stop, inclusive. A common error is miscounting the number of terms, so a sum from k = 3 to 7 has five terms, not four.
Key ideas
- Σ means sum
- 1 + 2 + … + n = n(n + 1)/2
- Geometric series can converge when ∣r∣ < 1
Quick check
Use cases: where Series and Sigma Notation helps
- Homework: review the key idea behind Series and Sigma Notation before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Series and Sigma Notation without notes.
- Visual learning: play with the interactive lab to see how Series and Sigma Notation changes when you adjust its inputs.
- Teaching a friend: explain Series and Sigma Notation in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Series and Sigma Notation?
A series adds up the terms of a sequence.
What will I learn in Series and Sigma Notation?
Σ means sum. 1 + 2 + … + n = n(n + 1)/2. Geometric series can converge when ∣r∣ < 1.
Is Series and Sigma Notation part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Series and Sigma Notation lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Series and Sigma Notation?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.