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10.7 Alternating Series Test for Convergence

1 min readjune 7, 2020

Welcome to AP Calc 10.7! In this lesson, you’ll how to test for convergence when dealing with an alternating series.


➕ Alternating Series Test Theorem

The alternating series test for convergence states that for an alternating series (1)nan\sum(-1)^n\cdot a_n, if

1.limnan=0 and1. \lim_{n\to \infty} a_n=0 \ \text{and}
2. an decreases,2. \ a_n \ \text{decreases,}

then the series converges. Otherwise, it diverges.

🧱 Breaking Down the Theorem

To illustrate this theorem, let’s look at a famous example—the alternating harmonic sequence:

n=1(1)nn\sum_{n=1}^\infty\frac{(-1)^n}{n}

Let’s look at our first criteria: the limit of ana_n must be equal to zero. To figure this out, we first must figure out what ana_n is! To do this, all we have to do is factor out the alternating part of the sequence, (1)n(-1)^n. Then, we get

an=1na_n=\frac{1}{n}

Now, let’s take the limit.

limn1n=0\lim_{n\to \infty}\frac{1}{n}=0

Well, that’s our first criteria satisfied! Now we need to know whether ana_n decreases. To show this, we must show that an>an+1a_n>a_{n+1}.

an=1n>1n+1=an+1a_n=\frac{1}{n}>\frac{1}{n+1}=a_{n+1}

Try plugging in a random number for nn to see that this is true!

a2=12>13=a2+1a_2=\frac{1}{2}>\frac{1}{3}=a_{2+1}

Therefore, both of our conditions for convergence are met, and our series converges!


📝 Practice with Alternating Series Test

Now it’s your turn to apply what you’ve learned!

❓Alternating Series Test Problems

For each of the following series, state whether they converge or diverge.

1.n=1(1)n+2n5n5+31. \sum_{n=1}^\infty \frac{(-1)^{n+2}\cdot n^5}{n^5+3}
2.n=2cos(nπ)n2. \sum_{n=2}^\infty\frac{\text{cos}(n\pi)}{n}
3.n=2(1)nln(n)3.\sum_{n=2}^\infty (-1)^n \cdot\text{ln}(n)

💡 Alternating Series Test Solutions

Question 1 Solution

First, identify ana_n.

an=n5n5+3a_n=\frac{n^5}{n^5+3}

Now, take the limit.

limnn5n5+3=10\lim_{n\to \infty}\frac{n^5}{n^5+3}=1\neq0

Since our first condition isn’t met, we don’t need to check the second condition. This series is divergent.

Question 2 Solution

This one is a little tricky—it requires you to recognize another type of alternating series, cos(nπ)\text{cos}(n\pi). If you plug some examples into your calculator, you’ll see that cos(nπ)=(1)n\text{cos}(n\pi)=(-1)^n. Therefore, we can treat this equation just like the harmonic series in our first example. We showed that the harmonic series met the conditions for convergence, so this one does too! This series is convergent.

Question 3 Solution

First, we find an=ln(n)a_n=\text{ln}(n). Then, we take the limit:

limnln(n)=\lim_{n\to \infty}\text{ln}(n)=\infty

Like our first problem, since the first condition isn’t met, we can say that this series is divergent without checking the other condition.


💫 Closing

Great work! With this test mastered, you’re well equipped to take on all sorts of convergence problems. Make sure you recognize both types of alternating series so that you know when to apply this test! 🧠

Key Terms to Review (1)

Alternating Sequences

: Alternating sequences are sequences in which the terms alternate between positive and negative values.

10.7 Alternating Series Test for Convergence

1 min readjune 7, 2020

Welcome to AP Calc 10.7! In this lesson, you’ll how to test for convergence when dealing with an alternating series.


➕ Alternating Series Test Theorem

The alternating series test for convergence states that for an alternating series (1)nan\sum(-1)^n\cdot a_n, if

1.limnan=0 and1. \lim_{n\to \infty} a_n=0 \ \text{and}
2. an decreases,2. \ a_n \ \text{decreases,}

then the series converges. Otherwise, it diverges.

🧱 Breaking Down the Theorem

To illustrate this theorem, let’s look at a famous example—the alternating harmonic sequence:

n=1(1)nn\sum_{n=1}^\infty\frac{(-1)^n}{n}

Let’s look at our first criteria: the limit of ana_n must be equal to zero. To figure this out, we first must figure out what ana_n is! To do this, all we have to do is factor out the alternating part of the sequence, (1)n(-1)^n. Then, we get

an=1na_n=\frac{1}{n}

Now, let’s take the limit.

limn1n=0\lim_{n\to \infty}\frac{1}{n}=0

Well, that’s our first criteria satisfied! Now we need to know whether ana_n decreases. To show this, we must show that an>an+1a_n>a_{n+1}.

an=1n>1n+1=an+1a_n=\frac{1}{n}>\frac{1}{n+1}=a_{n+1}

Try plugging in a random number for nn to see that this is true!

a2=12>13=a2+1a_2=\frac{1}{2}>\frac{1}{3}=a_{2+1}

Therefore, both of our conditions for convergence are met, and our series converges!


📝 Practice with Alternating Series Test

Now it’s your turn to apply what you’ve learned!

❓Alternating Series Test Problems

For each of the following series, state whether they converge or diverge.

1.n=1(1)n+2n5n5+31. \sum_{n=1}^\infty \frac{(-1)^{n+2}\cdot n^5}{n^5+3}
2.n=2cos(nπ)n2. \sum_{n=2}^\infty\frac{\text{cos}(n\pi)}{n}
3.n=2(1)nln(n)3.\sum_{n=2}^\infty (-1)^n \cdot\text{ln}(n)

💡 Alternating Series Test Solutions

Question 1 Solution

First, identify ana_n.

an=n5n5+3a_n=\frac{n^5}{n^5+3}

Now, take the limit.

limnn5n5+3=10\lim_{n\to \infty}\frac{n^5}{n^5+3}=1\neq0

Since our first condition isn’t met, we don’t need to check the second condition. This series is divergent.

Question 2 Solution

This one is a little tricky—it requires you to recognize another type of alternating series, cos(nπ)\text{cos}(n\pi). If you plug some examples into your calculator, you’ll see that cos(nπ)=(1)n\text{cos}(n\pi)=(-1)^n. Therefore, we can treat this equation just like the harmonic series in our first example. We showed that the harmonic series met the conditions for convergence, so this one does too! This series is convergent.

Question 3 Solution

First, we find an=ln(n)a_n=\text{ln}(n). Then, we take the limit:

limnln(n)=\lim_{n\to \infty}\text{ln}(n)=\infty

Like our first problem, since the first condition isn’t met, we can say that this series is divergent without checking the other condition.


💫 Closing

Great work! With this test mastered, you’re well equipped to take on all sorts of convergence problems. Make sure you recognize both types of alternating series so that you know when to apply this test! 🧠

Key Terms to Review (1)

Alternating Sequences

: Alternating sequences are sequences in which the terms alternate between positive and negative values.


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© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.