In the last few sets of supplemental notes, we have been investigating the convergence or divergence of positive term series. Now we are going to investigate the alternating series, and determine the convergence and divergence of these series.

FACT:

A series in which the terms are alternating between positive and negative is called an alternating series.

FACT:

converges if all three of the following conditions are satisfied:

1.

The u _n's are all positive.

2.

u _n ³ u _{n + 1} for all n ³ N, for some integer N.

3.

u _n ® 0

EXAMPLE 1:

Does the following series converge or diverge?

SOLUTION:

u _n ³ 0 for n ³ 1, so the first condition is satisfied. Now to see if the second condition holds. I will write out the first few terms of this sequence.

If I look at these terms u ₁ < u ₂ < u ₃ > u ₄ > . . . , but sometimes it not this easy to prove that the second condition holds. So I will plot the first 20 terms of this sequence. Here is the plot.

You should be able now to definitely say that u _n ³ u _{n + 1} for n ³ 3, so the second condition holds. Now to determine if the third condition holds.

The third condition is satisfied. Therefore this series converges by the alternating series test.

EXAMPLE 2:

Does the following series converge or diverge?

SOLUTION:

u _n ³ 0 for n ³ 1, so the first condition is satisfied. Now to see if the second condition holds. I will write out the first few terms of this sequence.

You should notice that u ₁ ³ u ₂ ³ u ₃ ³ . . . , so u _n ³ u _{n + 1} for all n ³ 1. Therefore, the second condition is satisfied. Now to determine if the third condition is satisfied.

The third condition is satisfied. Therefore this series converges by the alternating series test.

EXAMPLE 3:

Does the following series converge or diverge?

SOLUTION:

u _n ³ 0 for n ³ 1, so the first condition is satisfied. Now to see if the second condition holds. I will write out the first few terms of this sequence.

It looks like the second condition is going to hold, but I am going to plot the first 20 terms to make sure.

You should be able now to definitely say that u _n ³ u _{n + 1} for n ³ 1, so the second condition holds. Now to determine if the third condition holds.

The third condition is satisfied. Therefore this series converges by the alternating series test.

EXAMPLE 4:

Does the following series converge or diverge?

SOLUTION:

u _n ³ 0 for n ³ 1, so the first condition is satisfied. Now to see if the second condition holds. I will write out the first few terms of this sequence.

It looks like the second condition is going to hold, but I am going to plot the first 20 terms to make sure.

You should be able now to definitely say that u _n ³ u _{n + 1} for n ³ 1, so the second condition holds. Now to determine if the third condition holds.

The third condition does not hold because the limit of u _n does not go to zero. Therefore this series diverges by the alternating series test.

EXAMPLE 5:

Does the following series converge or diverge?

SOLUTION:

You should know by now that e ⁿ is always positive for n ³ 0, so u _n will be positive for n ³ 0. Therefore, the first condition is satisfied. Now to see if the second condition holds. I will write out the first few terms of this sequence.

You should notice that u ₁ ³ u ₂ ³ u ₃ ³ . . . , so u _n ³ u _{n + 1} for all n ³ 1. Therefore, the second condition is satisfied. Now to determine if the third condition is satisfied.

The third condition is satisfied. Therefore this series converges by the alternating series test.