So far, most of the series we studied consists of positive terms. In this lecture, we will consider more complicated situations where a series contains positive and negative terms. We will focus on the special case of alternating series whose convergence properties can be determined easily. We will see convergent alternating series converge in a very weak way which leads strange behaviors. To distinguish this and the stronger types of convergence, we introduce the concept of "absolute convergence".
Tianran Chen
Department of Mathematics
Auburn University at Montgomery
So far, most of the series we studied consists of positive terms. Such series converge or diverge in fairly simple ways. We will consider more complicated situations where a series contains positive and negative terms. We will focus on the special case of alternating series.
Definition. An alternating series is a series whose terms have alternating signs. It can be written in the form \[ \sum_{k=1}^\infty (-1)^k c_k \quad\text{or}\quad \sum_{k=1}^\infty (-1)^{k+1} c_k \quad \text{for positive } c_1,c_2,\ldots. \]
Here $(-1)^k$ and $(-1)^{k+1}$ are the common ways to express alternating signs (depending on if you start with $-1$ or $+1$).
\begin{align*} &1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots &\text{alternating series} \\ &1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots &\text{alternating series} \\ &1 - 1 + 1 - 1 + 1 - 1 + 1 - \cdots &\text{alternating series} \\ &\frac{\cos(\pi)}{1} + \frac{\cos(2\pi)}{2} + \frac{\cos(3\pi)}{3} + \cdots &\text{alternating series} \\ &1 - \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} - \frac{1}{6} + \cdots &\text{NOT alternating series} \end{align*}
Alternating series test. An alternating series of the form \[ \sum_{k=1}^\infty (-1)^k c_k \quad\text{or}\quad \sum_{k=1}^\infty (-1)^{k+1} c_k \quad \text{for positive } c_1,c_2,\ldots \] converges if both of the following conditions hold:
By this test, we can confirm the alternating harmonic series \[ \sum_{k=1}^\infty \frac{(-1)^{k+1}}{k} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} + \cdots \] is convergent (even though the harmonic series diverges).
The series \[ \sum_{k=1}^\infty \frac{(-1)^k}{\sqrt{k}} = -1 + \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} - \frac{1}{\sqrt{5}} + \cdots \] also satisfies the two conditions
Therefore, we can conclude that this series converges.
It is crucial to understand when not to use alternating series test. For the following alternating series, the conditions of the tests are not satisfies:
\[ \sum_{k=1}^\infty (-1)^{k+1} = 1 - 1 + 1 - 1 + 1 - 1 + 1 - \cdots \]
Or the series \[ \frac{1}{\sqrt{2}-1} - \frac{1}{\sqrt{2}+1} + \frac{1}{\sqrt{3}-1} - \frac{1}{\sqrt{3}+1} + \frac{1}{\sqrt{4}-1} - \cdots \]
Use the alternating series test (in combinations with other tests, if necessary), determine which of the following series is convergent.
The convergence of the alternating harmonic series is a weak form of convergence --- it converges only because of the cancellation between positive and negative terms. To further distinguish this and the more robust form of converges, we have the concept of "absolute convergence".
Definition. We say a series $\sum_{k=1}^\infty a_k$ converges absolutely (or it exhibits absolute convergence) if $\sum_{k=1}^\infty |a_k|$ converges. On the other hand, if $\sum_{k=1}^\infty a_k$ converges but $\sum_{k=1}^\infty |a_k|$ diverges, we say the series converges conditionally (or it exhibits conditional convergence).
From what we know about geometric series we can see the \[ \sum_{k=1}^\infty (-1)^{k+1} \left( \frac{1}{3} \right)^k = \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \cdots \] exhibits absolute convergence since \[ \sum_{k=1}^\infty \left| (-1)^{k+1} \left( \frac{1}{3} \right)^k \right| = \sum_{k=1}^\infty \left( \frac{1}{3} \right)^k \] converges.
It is important to understand that absolute convergence is a stronger condition than convergence, even though this is not clear from the definition.
Theorem. If a series converges absolutely, then it converges.
Absolute convergence is a stronger condition, and it is quite important in our study of series as it implies many nice properties: E.g. rearrangements of terms in an absolute convergent series do not change the value of the sum. (The same cannot be said about conditionally convergent series)
The alternating harmonic series is an example of conditionally convergent series: \[ \sum_{k=1}^\infty \frac{(-1)^{k+1}}{k} \quad\text{converges,} \] but \[ \sum_{k=1}^\infty \frac{1}{k} \quad\text{diverges.} \]
Determine which of the following alternating series is/are convergent.