Alternating Series

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An alternating series is a series whose terms are alternately positive and negative. Namely, it has the form

or where .

Here are two examples :

1. = + ... .

2. + ... .

3. = + ... .

For the first and second examples, since and , by The Test for Divergence , both series diverge.

Let's take a look at the third example = + ... .

Here we compute the nth partial sum for , 2, ... , 10 .

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We have better picture of the series from the plot below.

It seems that { } is decreasing and { } is increasing. It looks more clearly in the following plot, where { } is plotted with red circles and { } is plotted with blue circles.

We can see from the graph above that { } is increasing and bounded above by , while { } is decreasing and bounded below by ; by the monotonic Sequence Test, both { } and { } converge.

Moreover,

it follows that . Therefore, converges.

In general, we have the following theorem :

The Alternating Series Test

If and { } is decreasing with , then the alternating series

( or )

converges.

In fact,

= ( ) + ( ) + ... + ( )

since { } is decreasing, we have and { } is decreasing.

But we can also write

= - ... - ( ) -

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so for all n .

Therefore, by the Monotonic Sequence Theorem, { } converges, say .

Moreover, and , it follows that { } converges and

= S

Therefore, .

Note that (Why ?).

In this case,

= + ... + ....

For each , + ... + .... is also an alternating series which satisfies the assumption of the the Alternating Series Test. The following plot illustrates the case for ( the blue circle represents the point ( 41, ) ).

It is therefore convergent, let's call the limit , then we have

= .

We say the series is absolutely convergent if the series of absolute values converges.

For example, .

By the Integral Test, converges, so is absolutely convergent; and by the Alternating Series Test, we get that is convergent also.

Note that the series converges, but diverges. From this, we see that a convergent series may not be absolutely convergent.

Question :

Is it true that if is absolutely convergent then is convergent ?

Let's examine the difference between the convergence of the series and that of the series .

It seems converges much faster than .

Now let's compare the convergence of the series and .

The rate of convergence of these two series are about the same.

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From the graphs above, we see that the absolute value of the n th term of the series goes to zero more rapidly than that of .

It seems that we have a good reason to conjecture that if is absolutely convergent then is convergent.

Theorem If is absolutely convergent then is convergent.

Proof :

Note that + .

If is absolutely convergent then is convergent. So is convergent. Therefore, by Comparison Test,

is convergent.

Since , so is the difference of two convergent series and is therefore convergent.

We say a series converges conditionally iff converges but does not converge absolutely.

Now we have tests for positive terms and for alternating series. But what if the sign of the terms switches back and forth irregularly ? For example, ?

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We have no clue !!!

But it is rather different for the case if we consider the series .

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The series seems to converge !! We notice that the terms of the series converges to 0 much faster than that of the series and also converges absolutely (why ??).

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Something funny about the plot for --- It does not seem to be increasing ! Let's take a closer look.

It's O.K. It is increasing. But can we be sure that converge ?

Yes, since converges absolutely, by the theorem, converge.

Remark : If converges, by the theorem, converges. However, in general, . Examples ???

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Download Maple Worksheet

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