Alternating Series

TABLE OF CONTENTS :

TABLE OF CONTENTS

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Nie wieder prokastinieren mit unseren Lernerinnerungen.

When you pluck the string on an instrument, it moves to the left and right of the center, and eventually becomes still again. That is an example of an alternating harmonic series.

Alternating Series Formula

A series

$\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} a_{n}$

is called alternating if $a_{n} > 0$ are positive.

Is the series

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

an alternating series?

Answer: Yes this is an alternating series because you can write the series as

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

and let

$a_{n} = \frac{1}{n} > 0$ .

Remember that

$\sum_{n = 1}^{\infty} \frac{1}{n}$

is called the Harmonic series, so people call

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

the Alternating Harmonic series because it is based on the Harmonic series.

The Alternating Series Sum

You might be tempted to sum an alternating series by grouping sums like so:

$\sum_{n = 1}^{\infty} {(- 1)}^{n} = - 1 + 1 - 1 + 1 - 1 + 1 - \dots = (- 1 + 1) + (- 1 + 1) + (- 1 + 1) + \dots = 0 + 0 + 0 + 0 + \dots = 0,$

but let's look at the sequence of partial sums and see if that is true. The sequence of partial sums for this series is given by:

$\begin{matrix} s_{1} & = & - 1 \\ s_{2} & = & - 1 + 1 = 0 \\ s_{3} & = & 0 + - 1 = - 1 \\ s_{4} & = & - 1 + 1 = 0 \\ ⋮ \end{matrix}$

which as a sequence doesn't converge at all! Now you know the series doesn't converge, so the answer of 0 for the sum above isn't correct. This means that you can't group infinitely many terms using properties of addition like you would if it were a finite sum.

The Alternating Series Test

It would be nice to have a simple way to see if an alternating series will converge or not. That is where the Alternating Series Test comes in handy. This is also called Leibniz's Theorem.

Alternating Series Test (Leibniz's Theorem): If the alternating series

$\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} a_{n}$

has the properties that:

1. each $a_{n} > 0$ ;

2. $a_{n} \geq a_{n + 1}$ for all $n > N$ where $N$ is some fixed natural number; and

3. $\lim_{n \to \infty} a_{n} = 0$ ,

then the series converges.

You can also look at absolute versus conditional convergence for alternating series, and for more information on that see Absolute and Conditional Convergence

Going back to the Alternating Harmonic Series, you already know that the first condition of the Alternating Series Test is satisfied because

$a_{n} = \frac{1}{n} > 0$ .

Condition 2: Since

$a_{n} = \frac{1}{n}$ and $a_{n + 1} = \frac{1}{n + 1}$

you know that $a_{n} > a_{n + 1}$ because $n < n + 1$ , which implies that

$\frac{1}{n} > \frac{1}{n + 1}$ ,

so the second condition is satisfied.

Condition 3: You also know that

$\lim_{n \to \infty} a_{n} = \lim_{n \to \infty} \frac{1}{n} = 0$ ,

so the third condition is satisfied.

That means you can apply the Alternating Series Test to say that the Alternating Harmonic Series converges.

Remember that the Harmonic Series diverges, so sometimes changing something from a regular series to an alternating series can change it from diverging to converging.

It is very important to note that the Alternating Series Test can tell you if something converges, but it can't tell you if something diverges!

Alternating Series Examples

Let's look at some examples of alternating series, and see if they converge or not using the Alternating Series Test.

You know that the P-series

$\sum_{n = 1}^{\infty} {(\frac{1}{n})}^{2}$

converges because $p = 2$ .

What can you say about the Alternating P-series

$\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1}}{n^{2}}$ ?

Answer: Before applying the Alternating Series Test you need to be sure that all of the conditions are satisfied.

Condition 1: Here

$a_{n} = \frac{1}{n^{2}} > 0$

so the first condition is satisfied.

Condition 2: You know that

$a_{n} = \frac{1}{n^{2}}$

and

$a_{n + 1} = \frac{1}{{(n + 1)}^{2}} = \frac{1}{n^{2} + 2 n + 1} < \frac{1}{n^{2}} = a_{n}$ ,

so the second condition is satisfied.

Condition 3: Looking at the limit,

$\lim_{n \to \infty} a_{n} = \lim_{n \to \infty} \frac{1}{n^{2}} = 0$ ,

so the third condition is satisfied.

That means the Alternating Series Test tells you that the series

$\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1}}{n^{2}}$

converges.

Can you use the Alternating Series Test to tell if the series

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

converges?

Answer:

Let's check the conditions for the Alternating Series Test.

Condition 1: For this series

$a_{n} = \frac{1}{\sqrt{n}} > 0$ ,

so the first condition is satisfied.

Condition 2: Here

$a_{n} = \frac{1}{\sqrt{n}}$ and $a_{n + 1} = \frac{1}{\sqrt{n + 1}}$ .

You may not have a good intuition of how $a_{n}$ and $a_{n + 1}$ compare to each other, so let's try out a value for $n$ and see what happens. If $n = 10$ , then

$a_{10} = \frac{1}{\sqrt{10}} \approx 0.3162$ , and $a_{11} = \frac{1}{\sqrt{11}} \approx 0.3015$ ,

which means that $a_{10} > a_{11}$ . This is the direction you would hope the inequality would go. In fact, since $n < n + 1$ , you know that $\sqrt{n} < \sqrt{n + 1}$ which implies

$\frac{1}{\sqrt{n}} > \frac{1}{\sqrt{n + 1}}$ ,

so in general $a_{n} > a_{n + 1}$ . Now you know that the second condition holds.

Condition 3: Taking a look at the limit,

$\lim_{n \to \infty} a_{n} = \lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0$ ,

so the third condition holds as well.

Now by the Alternating Series Test, you know that the series converges.

Similar to the example earlier, you can show that the partial sums for the series

$\sum_{n = 1}^{\infty} {(- 1)}^{n + 1}$

diverges, which means the series diverges. What happens if you try and use the Alternating Series Test?

Answer:

First, let's look at the sequence of partial sums for this series. You know that

$\begin{matrix} s_{1} & = & 1 \\ s_{2} & = & 1 - 1 = 0 \\ s_{3} & = & 0 + 1 = 1 \\ s_{4} & = & 1 - 1 = 0 \\ ⋮ \end{matrix}$

so in fact the sequence of partial sums diverges, which by definition means the series diverges.

To apply the Alternating Series Test you need all three conditions satisfied. For this series, $a_{n} = 1$ for any $n$ , which means the first condition is satisfied. But when you check the second condition you need that $a_{n} > a_{n + 1}$ , or in other words, that $1 > 1$ which isn't true. It is even worse when you look at the limit in the third condition since

$\lim_{n \to \infty} a_{n} = \lim_{n \to \infty} 1 = 1$ ,

which is definitely not zero. So even though you know by looking at the partial sums that this series diverges, you can't use the Alternating Series Test to say anything about it.

You might want to use the logical equivalence "if A then B" is the same as "if B is false then A is false" to say that if one or more of the three conditions isn't satisfied, then the series diverges. However, it is incorrect to say that if one of the three conditions of the Alternating Series Test doesn't hold then the series diverges. Instead what you can say is that if an alternating series diverges, then it fails to satisfy one or more of the three conditions of the Alternating Series Test.

If you come across an alternating series where the third condition is false then you will want to try using the n^th Term Test for divergence instead. In fact, that is usually a good test to start with for alternating series since it is less work to apply than the Alternating Series Test. See Divergence Test for more details.

Look at the series

$\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1} n}{2 n - 1}$ .

This series diverges, which means that one or more of the three conditions of the Alternating Series Test fails. Which condition fails?

Answer:

Here

$a_{n} = \frac{n}{2 n - 1}$ ,

and you can see that the first condition holds.

Condition 2: To check condition 2, showing that $a_{n} > a_{n + 1}$ is the same as showing that $a_{n} - a_{n + 1} > 0$ . So

id="2905224" role="math" $a_{n} - a_{n + 1} = \frac{n}{2 n - 1} - \frac{n + 1}{2 (n + 1) - 1} = \frac{1}{(2 n - 1) (2 n + 1)} = \frac{1}{4 n^{2} - 1} > \frac{1}{4 n^{2}} > 0$

and the second condition holds.

Condition 3: Looking at the limit,

$\lim_{n \to \infty} \frac{n}{2 n - 1} = \frac{1}{2}$ ,

which is definitely not zero. That means condition 3 of the Alternating Series Test fails.

The series in the example above is shown to be divergent using the n^th Term Test for divergence. See Divergence Test for more details.

Alternating Series Estimation Theorem

Sometimes it is good enough to know approximately what an alternating series converges to, and how far off you are from the answer. For this, you can use the Alternating Series Bound theorem.

Theorem: Alternating Series Bound

If the alternating series

$\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} a_{n}$

has the properties that:

1. each $a_{n} > 0$ ;

2. $a_{n} \geq a_{n + 1}$ for all $n > N$ where $N$ is some fixed natural number; and

3. $\lim_{n \to \infty} a_{n} = 0$ ,

then the truncation error for the n^th partial sum is less than $a_{n + 1}$ and has the same sign as the first unused term.

Another way of talking about the truncation error is to remember that

$\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} a_{n}$

is actually the limit of the series, and

$\sum_{k = 1}^{n} {(- 1)}^{n + 1} a_{n}$

is the partial sum for the series, which is an approximation of what the limit is. The error is the difference between the limit and the partial sum, or in other words

ERROR $= |\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} a_{n} - \sum_{k = 1}^{n} {(- 1)}^{k + 1} a_{k}|$ .

The Alternating Series Bound tells you that

ERROR $= |\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} a_{n} - \sum_{k = 1}^{n} {(- 1)}^{k + 1} a_{k}| \leq a_{n + 1}$ .

Even better, you can tell if it is an overestimate or an underestimate by seeing if ${(- 1)}^{n + 1}$ is positive or negative. If it is positive then the partial sum is an overestimate, and if it is negative your partial sum is an underestimate.

Notice that these 3 conditions are exactly the same as for the Alternating Series Test! So if you can't apply the Alternating Series Test because one of the conditions isn't satisfied, you also can't apply the Alternating Series Bound theorem.

Let's take a look at the Alternating Harmonic series. You already know that the terms of the Alternating Series Bound theorem are satisfied from an earlier example. That means you are safe to apply the Alternating Series Bound theorem.

Suppose you add up the first 10 terms. How far off is the partial sum from the actual answer?

Answer: If you do the calculation,

$s_{10} = \frac{1267}{2520} \approx 0.6456$ ,

so using the Alternating Series Bound theorem the truncation error for $s_{10}$ is less than

id="2934012" role="math" $a_{11} = \frac{1}{11}$ .

The fact that the term ${(- 1)}^{11}$ is negative tells you that this is an underestimate rather than an overestimate.

Alternating Series - Key takeaways

A series
$\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} a_{n}$
is called alternating if $a_{n} > 0$ .
are positive.
Alternating Series Test (Leibniz's Theorem): If the alternating series
$\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} a_{n}$
has the properties that:
1. each $a_{n} > 0$ ;
2. $a_{n} \geq a_{n + 1}$ for all $n > N$ where $N$ is some fixed natural number; and
3. $\lim_{n \to \infty} a_{n} = 0$ ,
then the series converges.
Theorem: Alternating Series Bound
If the alternating series
$\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} a_{n}$
has the properties that:
1. each $a_{n} > 0$ ;
2. $a_{n} \geq a_{n + 1}$ for all $n > N$ where $N$ is some fixed natural number; and
3. $\lim_{n \to \infty} a_{n} = 0$ ,
then the truncation error for the n^th partial sum is less than $a_{n + 1}$ and has the same sign as the first unused term.

Frequently Asked Questions about Alternating Series

It is a series where it alternates between positive and negative terms.

It can be used to prove if an alternating series converges.

Well, you don't solve series, you solve equations. Are you looking to find the sum of an alternating series?

First, you can only use it on alternating series. Second, you need to know that the corresponding positive series has a sequence of decreasing terms which converge to 0.

You can approximate the sum of an alternating series by looking at the sequence of partial sums.

Flashcards in Alternating Series4 Start learning

What is another name for the Alternating Series Test?

Leibniz's Theorem.

Can the Alternating Series Test help you if one of the three conditions fails?

No. All the Alternating Series Test says is that if all three conditions are satisfied then the series converges. It doesn't say anything about divergence.

What test is good to use on an alternating series before you try the Alternating Series Test?

The n^th Term Test for divergence is a good one to use since it is less work to apply than the Alternating Series Test.

If you have an alternating series that you can't apply the Alternating Series Test to, can you use the Alternating Series Bound theorem?

No. The conditions to use either one are the same.

Learn with 4 Alternating Series flashcards in the free StudySmarter app

Already have an account? Log in

More about Alternating Series

Save Article

How would you like to learn this content?

Creating flashcards

Studying with content from your peer

Taking a short quiz

How would you like to learn this content?

Creating flashcards

Studying with content from your peer

Taking a short quiz

Free math cheat sheet!

Everything you need to know on . A perfect summary so you can easily remember everything.

Email Address*

Find Study Materials for

Create Study Materials

Alternating Series

Alternating Series Formula

The Alternating Series Sum

The Alternating Series Test

Alternating Series Examples

Alternating Series Estimation Theorem

Alternating Series - Key takeaways

Frequently Asked Questions about Alternating Series

Join over 22 million students in learning with our StudySmarter App

This is still free to read, it's not a paywall.

You need to register to keep reading

This is still free to read, it's not a paywall.

You need to register to keep reading

Entdecke Lernmaterial in der StudySmarter-App

Alternating Series

Alternating Series Formula

The Alternating Series Sum

The Alternating Series Test

Alternating Series Examples

Alternating Series Estimation Theorem

Alternating Series - Key takeaways

Frequently Asked Questions about Alternating Series

What is an alternating series?

What does the alternating series test prove?

How to solve alternating series?

When to use alternating series test?

How to approximate the sum of an alternating series?

Learn with 4 Alternating Series flashcards in the free StudySmarter app

Join over 22 million students in learning with our StudySmarter App

This is still free to read, it's not a paywall.

You need to register to keep reading

This is still free to read, it's not a paywall.

You need to register to keep reading

Create a free account to save this explanation.

Entdecke Lernmaterial in der StudySmarter-App

Join over 22 million students in learning with our StudySmarter App

Join over 22 million students in learning with our StudySmarter App