Determining Absolute or Conditional Convergence - AP Calculus BC Study Guide

The Core Idea: Determining Absolute or Conditional Convergence

When a series contains both positive and negative terms, its convergence can be classified more precisely. The fundamental question this topic addresses is whether the series converges because its terms are getting small fast enough on their own, or if it converges only due to the cancellation between its positive and negative terms. This distinction leads to two types of convergence for a series $\sum a_n$.

If the series formed by taking the absolute value of each term, $\sum |a_n|$, converges, then the original series is said to converge absolutely. This is a strong form of convergence, as it implies the original series $\sum a_n$ must also converge. However, it is possible for the original series $\sum a_n$ to converge while the series of absolute values $\sum |a_n|$ diverges. In this specific case, the series is said to converge conditionally, meaning its convergence is dependent on the alternating signs.

Key Definitions

The classification of convergence for a series $\sum a_n$ is based on the behavior of both the original series and the series of its absolute values, $\sum |a_n|$.

• Absolute Convergence: The series $\sum a_n$ is absolutely convergent if the series of absolute values, $\sum |a_n|$, converges.

• Conditional Convergence: The series $\sum a_n$ is conditionally convergent if the series $\sum a_n$ converges AND the series of absolute values, $\sum |a_n|$, diverges.

• Implication of Absolute Convergence: If the series $\sum |a_n|$ converges, then the original series $\sum a_n$ is guaranteed to converge. In other words, absolute convergence implies convergence.

Understanding the Hierarchy of Convergence

To determine the type of convergence for a series $\sum a_n$, there is a logical, two-step process. The key is to always begin by testing for absolute convergence, as it is the stronger condition.

Step 1: Test the series of absolute values, $\sum |a_n|$, for convergence.

Use any appropriate convergence test (e.g., p-series, geometric series, comparison tests, ratio test, integral test) on the series of positive terms $\sum |a_n|$.

• Path A: If $\sum |a_n|$ converges: The process is complete. By definition, the original series $\sum a_n$converges absolutely. Because absolute convergence implies convergence, you know $\sum a_n$ also converges.

• Path B: If $\sum |a_n|$ diverges: The work is not finished. You have only ruled out absolute convergence. You must proceed to Step 2 to determine if the series might converge conditionally or if it simply diverges.

Step 2 (Only if $\sum |a_n|$ diverges): Test the original series, $\sum a_n$, for convergence.

For series with alternating signs, this step typically involves the Alternating Series Test.

• If $\sum a_n$ converges: Since we know from Step 1 that $\sum |a_n|$ diverges, this outcome fits the definition of conditional convergence. The series $\sum a_n$converges conditionally.

• If $\sum a_n$ diverges: Since both the original series and the series of absolute values diverge, the series simply diverges.

Core Concepts & Rules

• A series $\sum a_n$ converges absolutely if the corresponding series of positive terms, $\sum |a_n|$, converges.

• If a series converges absolutely, it is guaranteed to converge.

• A series $\sum a_n$ converges conditionally if two conditions are met: (1) the series $\sum a_n$ converges, and (2) the series $\sum |a_n|$ diverges.

• The standard procedure is to first test for absolute convergence. If that test is conclusive (i.e., $\sum |a_n|$ converges), your work is done. If it is inconclusive (i.e., $\sum |a_n|$ diverges), you must then test the original series for convergence.

Step-by-Step Example 1: Conditional Convergence

Determine whether the series ${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{\sqrt{n}}$ converges absolutely, converges conditionally, or diverges.

Step 1: Test for absolute convergence.

We examine the series of the absolute values of the terms:

$$\sum _{n=1}^{\infty }\left|\frac{(-1{)}^{n+1}}{\sqrt{n}}\right|=\sum _{n=1}^{\infty }\frac{1}{\sqrt{n}}=\sum _{n=1}^{\infty }\frac{1}{n^{1/2}}$$

This is a p-series with $p=\frac{1}{2}$. Since $p\le 1$, the series ${\sum }_{n=1}^{\infty }\frac{1}{\sqrt{n}}$ diverges.

Therefore, the original series does not converge absolutely.

Step 2: Test the original series for convergence.

Since absolute convergence failed, we now test the original series, ${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{\sqrt{n}}$, using the Alternating Series Test.

Let $a_n=\frac{1}{\sqrt{n}}$.

1. Check if the terms are non-increasing: $a_{n+1}=\frac{1}{\sqrt{n+1}}\le \frac{1}{\sqrt{n}}=a_n$ for all $n\ge 1$. This is true.

2. Check if the limit of the terms is zero: ${\lim }_{n\to \infty }{a}_n={\lim }_{n\to \infty }\frac{1}{\sqrt{n}}=0$. This is true.

Since both conditions of the Alternating Series Test are met, the series ${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{\sqrt{n}}$ converges.

Step 3: Conclude.

The original series converges, but the series of absolute values diverges. By definition, the series ${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{\sqrt{n}}$converges conditionally.

Step-by-Step Example 2: Absolute Convergence

Determine whether the series ${\sum }_{n=1}^{\infty }\frac{\sin (n)}{n^2+1}$ converges absolutely, converges conditionally, or diverges.

Step 1: Test for absolute convergence.

We examine the series of the absolute values of the terms:

$$\sum _{n=1}^{\infty }\left|\frac{\sin (n)}{n^2+1}\right|=\sum _{n=1}^{\infty }\frac{|\sin (n)|}{n^2+1}$$

This is not a standard series type. We can use the Direct Comparison Test. We know that for all $n$, $0\le |\sin (n)|\le 1$. Therefore, we can establish an inequality:

$$\frac{|\sin (n)|}{n^2+1}\le \frac{1}{n^2+1}$$

Now, we must determine if the larger series, ${\sum }_{n=1}^{\infty }\frac{1}{n^2+1}$, converges. We can compare this series to the known convergent p-series ${\sum }_{n=1}^{\infty }\frac{1}{n^2}$ (where $p=2>1$). Since $\frac{1}{n^2+1}<\frac{1}{n^2}$ for all $n\ge 1$, and since ${\sum }_{n=1}^{\infty }\frac{1}{n^2}$ converges, the series ${\sum }_{n=1}^{\infty }\frac{1}{n^2+1}$ must also converge by the Direct Comparison Test.

Because the larger series ${\sum }_{n=1}^{\infty }\frac{1}{n^2+1}$ converges, our original series of absolute values, ${\sum }_{n=1}^{\infty }\frac{|\sin (n)|}{n^2+1}$, must also converge by the Direct Comparison Test.

Step 2: Conclude.

Since the series of absolute values, $\sum |a_n|$, converges, the original series ${\sum }_{n=1}^{\infty }\frac{\sin (n)}{n^2+1}$converges absolutely. There is no need to test the original series for convergence, as absolute convergence implies convergence.

Using Your Calculator

This topic is purely analytical. A graphing calculator cannot be used to determine or prove absolute or conditional convergence. The definitions rely on the behavior of infinite series, which a calculator can only approximate.

A calculator can be used to support your analytical conclusion or build intuition. For example, after analytically determining that $\sum \frac{(-1{)}^{n+1}}{n}$ converges conditionally, you could:

1. Calculate partial sums of the original series: $sum(seq((-1{)}^{(}N+1)/N,N,1,100))$ to observe that the sum appears to be approaching a limit (approx. 0.69).

2. Calculate partial sums of the absolute value series: $sum(seq(1/N,N,1,100))$ to observe that the sum is growing and not approaching a limit (approx. 5.18), which supports the conclusion that $\sum |a_n|$ diverges.

These calculations can help verify your work but are not a substitute for the rigorous justification required on the AP Exam.

AP Exam Quick Hit

Common Question Types

• Direct Classification: You are given a single alternating series and asked to classify it.

  • Example: "Does the series ${\sum }_{n=2}^{\infty }\frac{(-1{)}^n}{n\ln (n)}$ converge absolutely, converge conditionally, or diverge? Justify your answer."

• Multiple Choice Selection: You are presented with two or three series and asked to identify which ones fit a certain classification.

  • Example: "Which of the following series converge conditionally?

    I. $\sum \frac{(-1{)}^n}{n^2}$

    II. $\sum \frac{(-1{)}^n}{n}$

    III. $\sum \frac{(-1{)}^{n}n}{n+1}$"

• FRQ Justification: A part of a Free Response Question may ask you to determine the type of convergence for a series, often related to a Taylor series evaluated at the endpoint of its interval of convergence.

  • Example: "For the power series $f(x)={\sum }_{n=1}^{\infty }\frac{(x-3{)}^n}{n}$, the radius of convergence is $R=1$. Determine if the series converges absolutely, converges conditionally, or diverges at $x=2$."

Common Mistakes

• Stopping Too Early: A student tests $\sum |a_n|$, finds that it diverges, and incorrectly concludes that the original series $\sum a_n$ must also diverge. This mistake forgets the possibility of conditional convergence.

• Confusing Convergence Types: Stating that a series "converges" when the question specifically asks if the convergence is absolute or conditional. The distinction is critical.

• Mixing Up Definitions: Incorrectly stating that conditional convergence occurs when $\sum |a_n|$ converges but $\sum a_n$ diverges. This is impossible, as absolute convergence implies convergence.

• Forgetting Absolute Values in Ratio/Root Tests: When testing an alternating series for absolute convergence using the Ratio Test, students sometimes fail to apply the absolute value, which can lead to an incorrect limit and conclusion. For $\sum a_n$, the Ratio Test must be applied to $|a_n|$.