You might have been searching for “Alternate Series Test,” but the proper term is actually “Alternating Series Test.” This test is a useful tool in calculus for determining the convergence of certain types of infinite series, specifically those that alternate in sign. In this article, we’ll explore what the Alternating Series Test is and how it can be applied.
What is the Alternating Series Test?
The Alternating Series Test is used to determine whether an alternating series converges. An alternating series is one in which the signs of the terms alternate between positive and negative. A typical alternating series looks like this:
\[ \sum_{n=1}^{\infty} (-1)^{n-1} a_n = a_1 – a_2 + a_3 – a_4 + \dots \]
where \(a_n\) are the positive terms of the series.
For the Alternating Series Test to confirm that an alternating series converges, two key conditions must be met:\\
- Decreasing Terms: The sequence of terms \(a_n\) must be decreasing, meaning that \(a_{n+1} \leq a_n\) for all \(n\).
- Limit of Terms Approaching Zero: The limit of the terms \(a_n\) must approach zero as \(n\) approaches infinity:
\[ \lim_{n \to \infty} a_n = 0 \]
If both of these conditions are satisfied, the Alternating Series Test tells us that the series converges.
Why Does the Test Work?
The Alternating Series Test works because when the terms of the series decrease in size and approach zero, the positive and negative terms effectively cancel each other out over time. This results in the partial sums of the series becoming increasingly closer to a fixed value, meaning the series converges.
Example of the Alternating Series Test
Let’s apply the Alternating Series Test to the following series:
\[ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} = 1 – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + \dots \]
Here, \(a_n = \frac{1}{n}\).
- Decreasing Terms: The terms \(a_n = \frac{1}{n}\) form a decreasing sequence since \(\frac{1}{n+1} \leq \frac{1}{n}\).
- Limit of Terms Approaching Zero: The limit of \(a_n\) as \(n\) approaches infinity is:
\[ \lim_{n \to \infty} \frac{1}{n} = 0 \]
Since both conditions are satisfied, the series converges by the Alternating Series Test.
Conclusion
The Alternating Series Test is a powerful tool for determining the convergence of series that alternate in sign. By ensuring that the terms decrease in magnitude and approach zero, we can confidently say that the series converges. Remember to apply this test whenever you encounter an alternating series in your studies.
