Sequences and series
A list of numbers follows a rule. How do you reach any term without counting through the others, and what do the terms add up to?
Where you are 10 inside
What this area is about
A sequence is an ordered list of numbers a_1, a_2, a_3, \ldots made by a rule. This area asks two questions of it: what is the nth term, and what is the total of the first n terms, or of all of them?
How these fit together
Early patterns. Repeating patterns repeat a core; patterns from a rule are built from a start and a step, then studied for features the rule does not state.
Language. Sequences, terms and notation names the term a_n and its index n. Sigma notation writes a sum such as \sum_{k=1}^{n} a_k exactly.
Two families and two ways to define them. Arithmetic sequences add a common difference d, a linear rule; geometric sequences multiply by a common ratio r, an exponential one. Recursive formulas give each term from the one before plus a starting value.
Sums. Arithmetic series and partial sums total the first n terms. Geometric series total the first n terms of a geometric sequence, by a formula that needs r \ne 1. Infinite geometric series have a sum \displaystyle \frac{a_1}{1 - r} only when |r| < 1.
A path through it
The two early pattern lessons stand on their own. Then sequence basics, arithmetic sequences, and geometric sequences, which lead to recursive formulas. Sigma notation follows arithmetic sequences; from it come arithmetic partial sums, and with geometric sequences, finite geometric series, then infinite ones.
Next: series in calculus, proof by induction and exponential growth and decay.