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Learn Sequences and Series

Patterns of numbers, their sums and when infinite sums stay finite.

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A sequence is an ordered list of numbers following a rule, such as 3, 7, 11, 15, and a series is what you get by adding its terms. Arithmetic sequences add the same amount each time; geometric ones multiply by the same ratio. Their sums have neat formulas, like the trick that gives 1 + 2 + ... + 100 = 5050. The deeper question, central to calculus, is when an infinite series adds up to a finite number: 1 + 1/2 + 1/4 + ... equals 2, yet 1 + 1/2 + 1/3 + ... grows without bound. Learnr builds a sequences and series course at your level and teaches it in short interactive lessons.

How to learn Sequences and Series, step by step

Find the rule behind a pattern. For 3, 7, 11, 15, the difference is 4, so the nth term is 3 + 4(n - 1). For 2, 6, 18, 54, the ratio is 3, so it is 2 times 3 to the n - 1.
Derive the arithmetic sum yourself. Write 1 + 2 + ... + 100 forwards and backwards, add the two lines to get 100 copies of 101, then halve: 5050. That reasoning gives the general formula n(first + last)/2.
Master geometric sums and their limits. A finite geometric sum is a(1 - r to the n)/(1 - r). When the ratio is between -1 and 1, the infinite sum is a/(1 - r), so 1 + 1/2 + 1/4 + ... equals 2.
Learn the harmonic series as a warning. Group the terms of 1 + 1/2 + 1/3 + 1/4 + ... to show the sum passes every number. Terms shrinking to zero is necessary for convergence, but not enough.
Work through the convergence tests in order. Use the nth-term test, geometric and p-series facts (1/n to the p converges for p greater than 1), then comparison, ratio, integral and alternating series tests. Keep a one-page chart of when each applies.
Practice choosing tests a little each day. Picking the right test is the real skill. Short daily Learnr sessions with a streak help you see enough series that the right test starts to suggest itself.

What you'll learn

Write explicit and recursive formulas for sequences
Sum arithmetic and geometric series with formulas
Read and write sigma notation
Decide whether an infinite series converges using standard tests
Build Taylor and power series for common functions

What's inside your Sequences and Series course

A sample path Learnr builds for Sequences and Series. Your real course adapts to your goal and level.

1. Sequences, patterns and formulas
2. Arithmetic sequences and series
3. Geometric sequences and series
4. Sigma notation and infinite geometric series
5. Convergence tests
6. Power series and Taylor series

How long does it take to learn Sequences and Series?

Arithmetic and geometric sequences take about 2 to 4 weeks at 3 hours a week in an algebra course. Infinite series and convergence tests, usually in second-semester calculus, take another 5 to 8 weeks.

What you need before you start

Algebra and exponents for the first part. For infinite series and convergence tests, single-variable calculus including limits and integrals is needed.

Common mistakes to avoid

Concluding a series converges because its terms approach zero. The harmonic series has terms approaching zero but diverges.
Using the infinite geometric sum formula when the ratio is 1 or larger in size, where the series actually diverges.
Mixing up the sequence with the series: the terms can approach 0 while their running total grows forever.

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Sequences and Series questions, answered

What is the difference between a sequence and a series?

A sequence is a list of numbers, like 1, 1/2, 1/4. A series is the sum of those numbers, like 1 + 1/2 + 1/4. A sequence can approach a limit while its series does or does not.

How can an infinite sum be finite?

If the terms shrink fast enough, the running totals approach a fixed number. Walking half the remaining distance to a wall over and over covers a total distance equal to the original gap.