---
title: Series & Sequences
description: Sequences are ordered lists of numbers, while series are the sums of those lists. You will understand how to determine convergence or divergence, use various convergence tests, and represent functions as infinite power series, such as Taylor series.
category: mathematics
subcategory: calculus
difficulty: beginner, intermediate, advanced
url: /subject/series-and-sequences
---

# Series & Sequences

Sequences are ordered lists of numbers, while series are the sums of those lists. You will understand how to determine convergence or divergence, use various convergence tests, and represent functions as infinite power series, such as Taylor series.

## Available Resources

1 Videos • 1 Books • 1 Courses • 2 Websites

## Websites

### 1. Khan Academy Series & Sequences

Khan Academy's Calculus 2 unit on infinite series, combining short videos with graded practice. It covers convergence and divergence tests, geometric and alternating series, power series, and Taylor and Maclaurin polynomials, so learners can decide whether a series converges and build polynomial approximations of functions.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www.khanacademy.org/math/calculus-2/infinite-series

**Tags:** infinite-series, convergence-tests, power-series, taylor-series, calculus

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Videos

### 1. Essence of calculus, chapter 10 (Taylor series!)

A 3Blue1Brown animated video from the Essence of Calculus series showing how Taylor polynomials are built by matching a function's derivatives at a point. Viewers come away understanding why each term appears, how approximations improve with more terms, and what radius of convergence means.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www.youtube.com/watch?v=3d6DsjIBzJ4

**Tags:** taylor-series, polynomial-approximation, calculus, visual-intuition

## Courses

### 1. Calculus II: Sequences and Series

A Johns Hopkins course on Coursera covering sequences and series, Riemann sums, the definite integral, the Fundamental Theorem of Calculus, substitution and numerical integration. Learners finish able to approximate and compute integrals and apply them to data-driven modelling problems after a first differential calculus course.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.coursera.org/learn/calculus-through-data-and-modelling-series-and-integrals

**Tags:** courses, mathematics-statistics, calculus

## Books

### 1. Introduction to Real Analysis

**Author:** Robert G. Bartle, Donald R. Sherbert

A standard undergraduate textbook building real analysis from the completeness of the reals through sequences, limits, continuity, differentiation, and the Riemann integral. Its graded exercises train students to construct epsilon-delta proofs independently.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0471321486?tag=edmonddante07-20

**Tags:** real-analysis, epsilon-delta-proofs, sequences-and-series, proof-writing

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