L21: Introduction to Series

Lecture notes covering convergent and divergent sequences, and an introduction to series.

Summary

This lecture introduces the fundamental concepts of sequences and series. We explore the difference between convergent and divergent sequences based on their limits, and define what constitutes a series versus a sequence.

flowchart TD
    A([Sequences & Series]) --> B[Sequence<br/>a₁, a₂, a₃, ...]
    A --> C[Series<br/>a₁ + a₂ + a₃ + ...]
    B --> D{Limit as k→∞?}
    D -->|Finite| E["Convergent Sequence"]
    D -->|Infinite / DNE| F["Divergent Sequence"]
    C --> G[Finite Series<br/>Σₖ₌₁ⁿ aₖ]
    C --> H[Infinite Series<br/>Σₖ₌₁^∞ aₖ]
    E -.-> I["Convergent sequence is<br/>necessary (not sufficient)<br/>for convergent series"]
    F -.-> J["Divergent sequence →<br/>Divergent series"]

Sequence & Series

Convergent & Divergent Sequences

Two types of sequences are discussed in this lecture.

Example 1: Divergent Sequence

Consider the sequence: $$1, 3, 5, 7, \ldots$$

It is clear that the value of each successive term gets larger and larger.

The $k$th term is $a_k = 2k - 1$
As $k$ increases to infinity (i.e., $k \to \infty$), the value of $a_k$ cannot be determined

$$\lim_{k \to \infty} [a_k] = \lim_{k \to \infty} [2k - 1] = \infty$$

[!warning] Divergent Sequence Since the limit does not approach a finite value, the sequence is said to be divergent.

Example 2: Convergent Sequence

Consider the sequence: $$1.1, 1.01, 1.001, 1.0001, \ldots, 1 + \left(\frac{1}{10}\right)^k, \ldots$$

The $k$th term is $a_k = 1 + \left(\frac{1}{10}\right)^k$
As $k \to \infty$, $1 + \left(\frac{1}{10}\right)^k \to 1$

$$\lim_{k \to \infty} [a_k] = \lim_{k \to \infty} \left[1 + \left(\frac{1}{10}\right)^k\right] = 1$$

[!success] Convergent Sequence Therefore, this sequence converges to unity (converges to 1).

General Definition

[!important] Formal Definition In general, if the $k$th term of a sequence is $a_k$ and $\lim_{k \to \infty}[a_k]$ exists (i.e., approaches a finite value), the sequence is said to be convergent.

Series

[!note] What is a Series? A series is simply when the terms of a sequence are added:

$a_1, a_2, a_3, \ldots, a_n$ is a sequence

$a_1 + a_2 + a_3 + \ldots + a_n$ is a series

Types of Series

Type	Description	Notation
Finite Series	Sum of finite number of terms	$\sum_{k=1}^{n} a_k$
Infinite Series	Sum of infinitely many terms	$\sum_{k=1}^{\infty} a_k$

Summation Notation

The summation notation is $\Sigma$ (called sigma).

Examples

Series	Sigma Notation
$a_1 + a_2 + \ldots + a_n$	$\displaystyle\sum_{k=1}^{n} a_k$
$a_1 + a_2 + \ldots$ (infinite)	$\displaystyle\sum_{k=1}^{\infty} a_k$

Standard Formulas

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

Example Problems

Problem 1: Expressing Series Using Sigma Notation

Express the following series using sigma notation:

$$3 + 5 + 7 + 9 + 11$$

Solution:

This is an arithmetic sequence with first term 3 and common difference 2.

$$\sum_{k=1}^{5} (2k + 1)$$

Problem 2: Finding Limits of Sequences

Determine whether the sequence converges or diverges:

$$a_k = \frac{1}{k}$$