Defining Convergent and Divergent Infinite Series - AP Calculus BC Study Guide

The Core Idea: Defining Convergent and Divergent Infinite Series

In mathematics, we often deal with finite sums. This topic introduces a profound and counterintuitive idea: is it possible to add up an infinite number of terms and arrive at a single, finite value? An infinite series is precisely this concept—the sum of all the terms in an infinite sequence. For example, if we have a sequence of numbers $a_1,a_2,a_3,\dots$, the corresponding infinite series is the sum $a_1+a_2+a_3+\dots$, which is written in sigma notation as ${\sum }_{n=1}^{\infty }{a}_n$.

The fundamental question this topic addresses is how to determine if such an infinite sum has a meaningful, finite value. We investigate this by examining the behavior of its partial sums. A partial sum is the sum of a finite number of terms from the beginning of the series. By creating a sequence of these partial sums, we can observe their trend. If this sequence of partial sums approaches a specific, finite number as we include more and more terms, we say the infinite series converges to that number. If the sequence of partial sums grows without bound or fails to approach a single value, the series diverges.

Key Definitions

This topic is built upon a few precise definitions that connect the concepts of sequences, sums, and limits.

1. Infinite Series

An infinite series is the sum of the terms of an infinite sequence $\{a_n\}$. It is denoted by:

$$\sum _{n=1}^{\infty }{a}_n=a_1+a_2+a_3+\cdots +a_n+\dots$$

2. The n$th Partial Sum ($S_n)

The n$th partial sum, denoted $S_n, is the sum of the first $n$ terms of the series. It is a finite sum that "builds up" the infinite series.

$$S_n=\sum _{k=1}^n{a}_k=a_1+a_2+a_3+\cdots +a_n$$

3. Convergent Series

An infinite series ${\sum }_{n=1}^{\infty }{a}_n$ is said to converge to a sum $S$ if the sequence of its partial sums, $\{S_n\}$, converges to $S$. In the language of limits, this means:

$$\underset{n\to \infty }{\lim }{S}_n=S$$

where $S$ is a finite real number. The value $S$ is called the sum of the series.

4. Divergent Series

An infinite series ${\sum }_{n=1}^{\infty }{a}_n$ is said to diverge if its sequence of partial sums, $\{S_n\}$, diverges. This means that the limit ${\lim }_{n\to \infty }{S}_n$ either does not exist or is infinite. A divergent series does not have a finite sum.

Understanding the Sequence of Partial Sums

The most critical concept in this topic is that the convergence or divergence of an infinite series is determined entirely by the convergence or divergence of its corresponding sequence of partial sums.

It is essential to distinguish between two different sequences:

1. The sequence of terms, $\{a_n\}$: This is the list of individual numbers being added together: $a_1,a_2,a_3,\dots$.

2. The sequence of partial sums, $\{S_n\}$: This is the sequence of cumulative sums:

   • $S_1=a_1$

   • $S_2=a_1+a_2$

   • $S_3=a_1+a_2+a_3$

   • ...

   • $S_n=a_1+a_2+\cdots +a_n$

To determine if the series $\sum a_n$ converges, we do not look at the limit of the terms $a_n$. Instead, we must analyze the limit of the partial sums $S_n$. The entire question of series convergence boils down to a single question about a sequence: Does the sequence $\{S_n\}$ have a finite limit? If the answer is yes, the series converges to that limit. If the answer is no, the series diverges.

Core Concepts & Rules

• Series as a Sum: An infinite series, ${\sum }_{n=1}^{\infty }{a}_n$, represents the process of adding all the terms of an infinite sequence $\{a_n\}$.

• Partial Sums as Building Blocks: The n$th partial sum, $S_n = a_1 + a_2 + \dots + a_n, is a finite sum that approximates the total sum of the infinite series.

• The Definition of Convergence: A series converges if and only if its sequence of partial sums, $\{S_n\}$, converges to a finite limit.

• The Sum of a Series: If a series converges, its sum is defined as the limit of its sequence of partial sums: ${\sum }_{n=1}^{\infty }{a}_n={\lim }_{n\to \infty }{S}_n$.

• The Definition of Divergence: A series diverges if its sequence of partial sums fails to converge to a finite limit.

Step-by-Step Example 1: Finding the Sum from Partial Sums

Problem: The n$th partial sum of the series\sum_{n=1}^{\infty} a_n$ is given by the formula $S_n=\frac{5n^2-2}{2n^2+1}$. Determine whether the series converges or diverges. If it converges, find its sum.

Solution:

Step 1: Identify the given information.

We are given a formula for the sequence of partial sums, $\{S_n\}$.

$$S_n=\frac{5n^2-2}{2n^2+1}$$

Step 2: Apply the definition of convergence.

According to the definition, the series $\sum a_n$ converges if the limit of its sequence of partial sums, ${\lim }_{n\to \infty }{S}_n$, exists and is finite. The sum of the series is this limit.

Step 3: Calculate the limit of the sequence of partial sums.

We need to evaluate ${\lim }_{n\to \infty }{S}_n$. Since this is a limit at infinity of a rational function of $n$, we can divide the numerator and denominator by the highest power of $n$, which is $n^2$.

$$\underset{n\to \infty }{\lim }{S}_n=\underset{n\to \infty }{\lim }\frac{5n^2-2}{2n^2+1}$$

$$\underset{n\to \infty }{\lim }\frac{\frac{5n^2-2}{n^2}}{\frac{2n^2+1}{n^2}}=\underset{n\to \infty }{\lim }\frac{5-\frac{2}{n^2}}{2+\frac{1}{n^2}}$$

As $n\to \infty$, the terms $\frac{2}{n^2}$ and $\frac{1}{n^2}$ both approach 0.

$$=\frac{5-0}{2+0}=\frac{5}{2}$$

Step 4: State the conclusion.

The limit of the sequence of partial sums is a finite number, $\frac{5}{2}$. Therefore, the series ${\sum }_{n=1}^{\infty }{a}_n$ converges. The sum of the series is the value of the limit.

Final Answer: The series converges, and its sum is $\frac{5}{2}$.

Step-by-Step Example 2: Deducing Convergence from a Table

Problem: The first five partial sums of the series ${\sum }_{k=1}^{\infty }{b}_k$ are given in the table below.

Does the series $\sum b_k$ appear to converge or diverge? If it appears to converge, to what value? Justify your reasoning.

Solution:

Step 1: Analyze the provided data.

The table gives us the values of the sequence of partial sums $\{S_n\}$ for $n=1,2,3,4,5$. We are asked to determine the behavior of the series by observing the trend in these partial sums.

Step 2: Observe the trend in the sequence of partial sums, $\{S_n\}$.

Let's examine the values:

• $S_1=0.66667$

• $S_2=0.83333$

• $S_3=0.90000$

• $S_4=0.93333$

• $S_5=0.95238$

The values of $S_n$ are increasing as $n$ increases. However, the amount they increase by seems to be getting smaller. Let's look at the differences:

• $S_2-S_1\approx 0.16666$

• $S_3-S_2\approx 0.06667$

• $S_4-S_3\approx 0.03333$

• $S_5-S_4\approx 0.01905$

The partial sums are increasing, but they appear to be approaching a ceiling or a horizontal asymptote. The values seem to be getting closer and closer to 1.

Step 3: Formulate a conjecture based on the definition of convergence.

The definition of a convergent series is that its sequence of partial sums, $\{S_n\}$, approaches a finite limit. Based on the trend in the table, it is reasonable to conjecture that ${\lim }_{n\to \infty }{S}_n=1$.

Step 4: State the conclusion with justification.

Because the sequence of partial sums $\{S_n\}$ appears to be approaching the finite value of 1, the series $\sum b_k$ appears to converge. The sum of the series would be the value of this limit.

Final Answer: The series appears to converge to a sum of 1. This is because the sequence of partial sums $\{S_n\}$, as shown in the table, is increasing and appears to be approaching a finite limit of 1.

Using Your Calculator

A graphing calculator cannot definitively prove that a series converges, as it cannot compute a limit to infinity. However, it is an excellent tool for investigating the behavior of the sequence of partial sums $\{S_n\}$ to form a conjecture.

Problem: Investigate the convergence of the series ${\sum }_{n=1}^{\infty }\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$.

Actionable Steps (TI-84 Style):

1. Define the terms of the series:

   • Press Y=.

   • In Y1, enter the formula for the terms of the series, using $X$ for n`: `Y1 = 1/√(X) - 1/√(X+1)`. 2. **Calculate a sequence of partial sums:** * Go to the home screen. * To calculate $S_{10}, use the $sum$ and $seq$ commands. Press 2ndSTAT$►$MATH5:sum(. Then press 2ndSTAT$►$OPS` `5:seq(`. * The syntax is $sum(seq(expression, variable, start, end)).

   • Enter: sum(seq(Y1, X, 1, 10)) and press ENTER. This calculates $S_{10}\approx 0.6838$.

   • Repeat for larger values of $n$:

     • sum(seq(Y1, X, 1, 100)) gives $S_{100}\approx 0.9005$.

     • sum(seq(Y1, X, 1, 1000)) gives $S_{1000}\approx 0.9685$.

     • sum(seq(Y1, X, 1, 10000)) gives $S_{10000}\approx 0.9901$.

2. Analyze the results and form a conjecture:

   • The sequence of partial sums $\{S_n\}$ appears to be increasing and approaching a value very close to 1.

   • Based on this numerical evidence, we can conjecture that the series converges to a sum of 1.

Important Note: This calculator process does not constitute a mathematical proof. It is a method for numerical investigation. A formal proof would require finding an explicit formula for $S_n$ and taking its limit analytically.

AP Exam Quick Hit

Common Question Types

• Given an explicit formula for $S_n$: You will be given a formula for the n$th partial sum, such as $S_n = \frac{2n}{3n+5}, and asked to determine if the series converges and find its sum. The solution is to simply compute ${\lim }_{n\to \infty }{S}_n$.

• Given a table of values for $S_n$: A table showing $n$ and the corresponding $S_n$ will be provided. You will be asked to determine if the series appears to converge or diverge based on the trend in the table and to justify your conclusion by referencing the limit of the partial sums.

• Conceptual Questions: A multiple-choice question might ask for the definition of a convergent series, requiring you to identify the correct statement involving the limit of the sequence of partial sums. For example: "Which of the following guarantees that the series $\sum a_n$ converges to a sum $S$?" The correct answer would be ${\lim }_{n\to \infty }{S}_n=S$, where $S_n={\sum }_{k=1}^n{a}_k$.

Common Mistakes

• Confusing the Sequence of Terms with the Sequence of Partial Sums: The most common error is to analyze the limit of the terms, ${\lim }_{n\to \infty }{a}_n$, instead of the limit of the partial sums, ${\lim }_{n\to \infty }{S}_n$. The definition of convergence for this topic is exclusively about the behavior of $\{S_n\}$.

• Incorrect Justification: Stating that a series converges because the partial sums "are getting closer to a number" is not sufficient. The formal justification must involve the concept of a limit: "The series converges because the limit of the sequence of its partial sums, ${\lim }_{n\to \infty }{S}_n$, is a finite number."

• Assuming a Pattern: When given a few terms of a series, students might incorrectly find a formula for $S_n$. Unless the series has a clear structure (like a telescoping series, though it won't be named), you are typically not expected to find a formula for $S_n$ from scratch; it will be given to you or presented in a table.