11.2 Series: Infinite Series, Geometric Series, and Convergence

11.2 Series and Summation Notation Notes

Objectives:

1. Define Infinite Series

2. Evaluate Geometric Series

3. Test for Divergence

Series

A series is the sum of a sequence.

• Finite Series: The sum of a finite number of terms from a sequence.

• Infinite Series: The sum of infinitely many terms from a sequence.

Summation notation

A summation notation is used to represent series compactly.

1. Finite Series: $S_n = \sum_{i=1}^n a_i = a_1 + a_2 + a_3 + \cdots + a_n$

2. Infinite Series: $S_n = \sum_{i=1}^{\infty} a_i = a_1 + a_2 + a_3 + \cdots$

Convergence and Divergence of Series

Partial Sums and Definitions

Given a series $\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots$, the nth partial sum is:

• $s_n = \sum_{i=1}^n a_i = a_1 + a_2 + \cdots + a_n$

If the sequence of partial sums $\{s_n\}$ converges and $\lim_{n \to \infty} s_n = s$ exists as a real number, then the series is convergent and $s$ is called the sum of the series. If $\{s_n\}$ diverges, the series is divergent.

Infinite Series Examples

1. Infinite Series for $\sin(1)$

2. Some functions can be represented as infinite series. For example, the Maclaurin series for $\sin(x)$ is:

   $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \frac{x^{11}}{11!} + \cdots$

   for $x = 1$:

Zeno's Paradox

Example of Zeno's Paradox involves evaluating the sum:

$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\cdots+\frac{1}{2^{n}}=\sum_{n=0}^{\infty}\frac{1}{2^n}$

This is a geometric series with first term $a = 1$ and ratio $r = \frac{1}{2}$.

Evaluating Geometric Series

• A geometric series is a series of the form:

• $\sum_{n=1}^{\infty} a_1 r^{n-1} = a_1 + a_1 r + a_1 r^2 + \cdots$

Let $S_{n}$denote its nth partial sum:

• $S_{n}=a_1+a_1r+a_1r^2+\cdots+a_1r^{n-1}$

• $S_n = a_1 \frac{1 - r^n}{1 - r}$, for $r \neq 1$

If $|r| < 1$, the infinite geometric series converges to:

$S = \frac{a_1}{1 - r}$

Examples from Notes 11.2

3. Find the sum of the geometric series: $5-\frac{10}{3}+\frac{20}{9}-\frac{40}{27}+\cdots$

• Find the sum of the Geometric Series: $\sum_{n=4}^{\infty} 7 \left(-\frac{3}{4}\right)^{n-1}$

4. Find the sum of the Geometric Series: $\sum_{n=1}^{\infty}2^{2n}3^{1-n}$

Convergence Tests

Test for Divergence

To determine if a series $\sum_{n=1}^{\infty} a_n$ converges or diverges, use the following test:

1. If the limit as n approaches infinity equals to zero or DOES NOT EXIST, then the series diverges.

2. If $\lim_{n\to\infty}a_{n}\ne0$, then it is considered INCONCLUSIVE and further tests are needed to determine its convergence.

Theorem

• If the series $\sum_{n=1}^{\infty} a_n$ is convergent, then $\lim_{n \to \infty} a_n = 0$

• Example from 11.3 Notes: Determine if the series converges or diverges for:

Properties of Convergent Series

Algebraic Properties

If $\sum a_n$ and $\sum b_n$ are convergent series, then:

Property	Formula
Sum	$\sum (a_n + b_n) = \sum a_n + \sum b_n$
Difference	$\sum (a_n - b_n) = \sum a_n - \sum b_n$
Constant Multiple	$\sum (c a_n) = c \sum a_n$

Examples

Convergence and Sum

1. Show that $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ converges and find its sum.

2. Determine if $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ converges or diverges.

3. Determine if $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n+1}$ converges or diverges.

4. Find the sum of $\sum_{n=1}^{\infty}\left(\frac{3}{n(n+1)}+\frac{1}{2^n}\right)$

Summary Table: Series Types and Convergence

Series Type	General Form	Convergence Criteria
Geometric Series	$\sum_{n=1}^{\infty} a_1 r^{n-1}$	$|r| < 1$
p-Series	$\sum_{n=1}^{\infty} \frac{1}{n^p}$	$p > 1$
Alternating Series	$\sum_{n=1}^{\infty} (-1)^n a_n$	Test for absolute/conditional convergence

Additional info:

• Some examples and exercises are provided for practice, such as evaluating the convergence and sum of specific series.

• Definitions and theorems are included to clarify the concepts of convergence and divergence.

• Geometric series and their sums are emphasized, including applications to paradoxes and function expansions.