Series

A series is the sum of the terms of a sequence.

Definition

If $a_1, a_2, a_3, \ldots, a_n, \ldots$ is a sequence, then:

$$a_1 + a_2 + a_3 + \cdots + a_n + \cdots$$

is called a series.

[!note] Key Distinction

  • Sequence: $a_1, a_2, a_3, \ldots$ (list of terms)
  • Series: $a_1 + a_2 + a_3 + \ldots$ (sum of terms)

Types of Series

Finite Series

A series with a finite number of terms.

$$S_n = a_1 + a_2 + a_3 + \cdots + a_n = \sum_{k=1}^{n} a_k$$

This is also called a partial sum of an infinite series.

Infinite Series

A series with infinitely many terms.

$$S = a_1 + a_2 + a_3 + \cdots = \sum_{k=1}^{\infty} a_k$$

Summation Notation

The sigma notation $\Sigma$ provides a compact way to write series.

Basic Notation

$$\sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \cdots + a_n$$

Component Meaning
$\Sigma$ Summation symbol (Greek letter sigma)
$k=1$ Lower limit (starting index)
$n$ Upper limit (ending index)
$a_k$ General term
$k$ Index of summation

Examples

Series Sigma Notation
$1 + 2 + 3 + \cdots + n$ $\displaystyle\sum_{k=1}^{n} k$
$1 + 4 + 9 + 16 + \cdots + 100$ $\displaystyle\sum_{k=1}^{10} k^2$
$a_1 + a_2 + a_3 + \cdots$ $\displaystyle\sum_{k=1}^{\infty} a_k$

Standard Summation Formulas

Constant Series

$$\sum_{k=1}^{n} c = nc \quad \text{(where $c$ is a constant)}$$

Arithmetic Series

$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$$

$$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$

$$\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2$$

Geometric Series

$$\sum_{k=0}^{n} r^k = \frac{1-r^{n+1}}{1-r} \quad \text{(for $r \neq 1$)}$$

For infinite geometric series with $|r| < 1$:

$$\sum_{k=0}^{\infty} r^k = \frac{1}{1-r}$$

Convergence of Series

Convergent Series

An infinite series $\sum_{k=1}^{\infty} a_k$ is convergent if the sequence of partial sums converges to a finite limit.

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

Divergent Series

If the sequence of partial sums does not approach a finite limit, the series is divergent.

Convergence Tests Flowchart

The following flowchart outlines the common convergence tests used to determine whether an infinite series converges or diverges:

graph TB
    Start("Start: Infinite Series") --> CheckType{"Terms are all positive?"}
    
    CheckType -->|Yes| Pos["Positive-Term Tests"]
    CheckType -->|No| CheckAlt{"Alternating signs?"}
    
    CheckAlt -->|Yes| Alt["Alternating Series Test"]
    CheckAlt -->|No| Other["Other Specialized Tests"]
    
    Pos --> Comp["Comparison Test"]
    Pos --> Ratio["Ratio Test"]
    Pos --> Root["Root Test"]
    Pos --> Int["Integral Test"]
    
    Comp --> Res("Convergent or Divergent")
    Ratio --> Res
    Root --> Res
    Int --> Res
    Alt --> Res
    Other --> Res
    
    style Start fill:#e7f5ff,stroke:#1971c2
    style Res fill:#d3f9d8,stroke:#2f9e44

Related Concepts

Concept Map

mindmap
  root((From Sequences to Series))
    Sequences
      Ordered lists of numbers
      Convergence via limits
      Arithmetic and geometric
    Series
      Sum of sequence terms
      Partial sums
      Convergence tests
      Power Series
        Infinite degree polynomials
        Radius and interval
        Taylor Series
          Approximation around a
        Maclaurin Series
          Special case at 0