State only the supported conclusion

Three series, three results

The teal dots are the partial sums of an alternating series, joined by a zigzag line. The orange bars are the sizes, which are the terms without their signs.

Series A has sizes \(\frac{1}{(n+1)^2}\), series B has sizes \(\frac{1}{(n+1)^{1/3}}\), and series C has sizes \(\frac{3n+1}{2n}\).

The list names the conditions of the alternating series test: the sizes are positive, fall from some term on and approach 0. A green check means that a condition holds, and a red cross means that it fails.

Press the buttons for series A and B. All three conditions hold, and the zigzag narrows toward one value as the slider adds terms.

For series C, the sizes approach \(\frac32\), so the third condition fails and the test gives no conclusion. The series still diverges, but for another reason: its terms do not approach 0.

What each result supports

Press each button to break one condition. A red bar is a negative size.

If all three conditions hold, the sentence under the list says that the series converges, the only conclusion that the test gives.

If any condition fails, the test shows neither convergence nor divergence, so you need another argument.

Even when the series converges, the test does not give its sum.

Notation

1. Let \(a_n\) be the size of the nth term. If \(a_n\) is positive, falls from some term on and approaches 0, then the series converges.

   \[\begin{aligned}&\cy{a_n}>0\\&\cy{a_{n+1}}\le\cy{a_n}\ \text{from some }n\text{ on}\\&\lim_{n\to\infty}\cy{a_n}=0\\&\Rightarrow\ \sum(-1)^n\cy{a_n}\ \text{converges}\end{aligned}\]

2. Convergence is the only conclusion of this test, so it never shows that a series diverges.

3. If the sizes do not approach 0, this test gives no conclusion. The series diverges by the nth-term test, because its terms do not approach 0.

4. The test shows that the partial sums approach some number, but it does not give that number.

When it fails

When a condition fails, the series can converge or diverge. The sizes \(\frac{2+(-1)^n}{n}\) rise at every even term, so the second condition fails. Each term is \(\frac{2(-1)^n}{n}+\frac1n\), and the first parts have a convergent sum by this test, while \(\sum\frac1n\) diverges, so this series diverges.

The sizes \(\frac{2+(-1)^n}{n^2}\) fail the same condition, but each term is \(\frac{2(-1)^n}{n^2}+\frac1{n^2}\), and both parts have convergent sums, so this series converges.

Example 1: All three conditions hold

What does the alternating series test show for \(\displaystyle\sum_{n=1}^{\infty}(-1)^n\frac{4}{5n+2}\)?

1. The sizes \(a_n=\frac{4}{5n+2}\) are positive for \(n\ge1\), because the top and the bottom are positive.

2. The top is fixed, and the bottom grows with n, so each size is smaller than the one before.

   \[\begin{aligned}(5(n+1)+2)-(5n+2)&=5\\&>0\end{aligned}\]

3. The bottom grows without bound, so the sizes approach 0.

   \[\lim_{n\to\infty}\frac{4}{5n+2}=0\]

4. All three conditions hold, so the series converges. This is all the test shows.

Example 2: A condition fails

What can you say about \(\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2n}{n+3}\)?

1. The sizes are positive, but they rise toward the dashed line at 2, not toward 0, so the second and third conditions fail. The limit comes from dividing the top and the bottom by \(n\).

   \[\lim_{n\to\infty}\frac{2n}{n+3}=\lim_{n\to\infty}\frac{2}{1+3/n}=2\]

2. A tempting wrong move: The third condition fails, so the alternating series test shows that the series diverges.

   A condition fails, so the sentence under the list reads "no conclusion from this test". This test can show only convergence.

3. The terms do not approach 0, because their sizes approach 2. The series diverges by the nth-term test, not by the alternating series test.

Practice

What does the alternating series test show for \(\displaystyle\sum_{n=1}^{\infty}(-1)^n\frac{n+2}{n^2+5}\)? State only the conclusion that the test supports.

The orange bars are the sizes. State a conclusion only if they are positive, fall and approach 0.

Questions

The graph shows two alternating series. Series X has sizes \(\frac{3}{n+1}\), and series Y has sizes \(\frac{n+1}{n}\). For each series, the teal dots are the partial sums and the orange bars are the sizes.

1. What does the alternating series test show for series X?

   • The series converges.
   • The series converges to 1.5.
   • The test gives no conclusion.
   Answer

   The series converges.

2. What can you say about series Y?

   • The test gives no conclusion. The terms do not approach 0, so the series diverges by the nth-term test.
   • The series converges, because the bars fall.
   • The alternating series test shows that the series diverges.
   • Nothing can be said about the series.
   Answer

   The test gives no conclusion. The terms do not approach 0, so the series diverges by the nth-term test.

3. Which statement does the alternating series test support for \(\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{1}{\sqrt n+2}\)?

   • The series converges.
   • The series diverges, because the sum of the sizes diverges.
   • The series converges to 0.
   • The test gives no conclusion.
   Answer

   The series converges.

Summary

You state that the series converges when the sizes are positive, fall from some term on and approach 0, and you state nothing more.

This works because these conditions make the zigzag of partial sums narrow to one value, but they do not give that value.

When a condition fails, you use another test, such as the nth-term test.

This is one of four skills in topic 10.7 of Which & Why: AP Calculus.