Sequences and Series — Mathematics Knowledge System

Overview

A sequence is an ordered list of numbers, a₁, a₂, a₃, ..., each indexed by a positive integer — formally, a function from the natural numbers to the reals. A series is the sum of the terms of a sequence, S = a₁ + a₂ + a₃ + .... Sequences describe discrete processes step by step (compound interest each period, populations each generation), while series answer the natural follow-up question: what do you get when you add all those steps together, and does that sum settle down to a finite value or grow without bound?

Intuition

Think of a sequence as a numbered list of stops along a path, and a series as the total distance traveled after visiting each stop in order. Some series behave like walking toward a wall you never quite reach — the partial sums creep toward a fixed number and converge — while others behave like walking in a straight line forever, growing without bound (diverging). Which behavior happens depends entirely on how fast the underlying sequence's terms shrink.

Interactive Graph

Formal Definition

Definition

A sequence is a function a: ℕ → ℝ, usually written aₙ. Its associated series is the sum of its terms, tracked via partial sums:

\{a_n\}_{n=1}^{\infty} = a_1, a_2, a_3, \ldots

Sequence

S_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n

Partial sum

S = \sum_{k=1}^{\infty} a_k = \lim_{n\to\infty} S_n

The series converges if this limit exists (is finite); otherwise it diverges

Infinite series

Notation

Notation	Meaning
$a_n$	The n-th term of the sequence
$\sum_{k=1}^{n} a_k$	Summation notation — the sum of terms from k=1 to n
$S_n$	The n-th partial sum — the sum of the first n terms

Properties

Recursive definition

a_n = f(a_{n-1}, a_{n-2}, \ldots) \text{ with given initial term(s)}

Explicit (closed) form

a_n = g(n) \text{ directly, no reference to prior terms}

Convergence requires terms → 0

\text{If } \sum a_k \text{ converges, then } \lim_{n\to\infty} a_n = 0 \text{ (necessary but not sufficient)}

Linearity of series

\sum (a_k + b_k) = \sum a_k + \sum b_k, \quad \sum c\,a_k = c\sum a_k

Applications

Recursive algorithms and their runtime analyses are often expressed and solved as recurrence relations, a form of sequence definition.

Worked Examples

Substitute n = 1, 2, 3, 4, 5 into the explicit formula.
$a_1=1,\ a_2=3,\ a_3=5,\ a_4=7,\ a_5=9$

Answer: 1, 3, 5, 7, 9 (the odd numbers)

Practice Problems

Difficulty 3/10

Write the first 4 terms of the recursively defined sequence a₁=3, aₙ=aₙ₋₁+5.

Difficulty 4/10

Find the 3rd partial sum of the series with terms aₙ = n².

Common Mistakes

Common Mistake

Assuming that if the terms of a sequence go to 0, the series must converge.

Terms going to 0 is necessary but NOT sufficient. The harmonic series Σ(1/n) has terms → 0 but its partial sums grow without bound (it diverges) — a classic counterexample.

Common Mistake

Confusing a sequence (an ordered list of terms) with a series (the sum of those terms).

A sequence {aₙ} is just the list 1, 4, 9, 16, .... The related series Σaₙ is the running total 1, 5, 14, 30, .... They are related but distinct objects with different notation.

Summary

A sequence {aₙ} is an ordered, indexed list of numbers; it can be defined explicitly (aₙ=g(n)) or recursively.
A series Σaₙ is the sum of a sequence's terms, tracked via partial sums Sₙ.
An infinite series converges if the sequence of partial sums Sₙ approaches a finite limit; otherwise it diverges.
Terms approaching 0 is necessary but not sufficient for convergence (e.g. the harmonic series diverges).
Sequences and series underlie recurrence relations, annuities, and discrete models of growth and decay.

References

BookSullivan, M. Algebra and Trigonometry, 10th ed. Ch. 12 — Sequences, Induction, and Probability.
WebsiteWikipedia — Sequence