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4 Sep 2024
The Alternating Series Test is a fundamental tool in calculus used to determine whether certain infinite series converge. This test specifically applies to series that alternate in sign, meaning the terms switch between positive and negative. In this article, we’ll discuss the test in detail and walk through three examples of increasing difficulty.
The Alternating Series Test applies to a series of the form:
\[ \sum_{n=1}^{\infty} (-1)^{n-1} a_n = a_1 – a_2 + a_3 – a_4 + \dots \]
where \(a_n > 0\). The test states that the series converges if two conditions are met:
\[ \lim_{n \to \infty} a_n = 0 \]
If both conditions hold, the series converges.
Consider the series:
\[ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} = 1 – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + \dots \]
Here, \(a_n = \frac{1}{n}\).
\[ \lim_{n \to \infty} \frac{1}{n} = 0 \]
Since both conditions are satisfied, the series converges by the Alternating Series Test. This is known as the alternating harmonic series.
Consider the series:
\[ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2} = 1 – \frac{1}{4} + \frac{1}{9} – \frac{1}{16} + \dots \]
Here, \(a_n = \frac{1}{n^2}\).
\[ \lim_{n \to \infty} \frac{1}{n^2} = 0 \]
Again, both conditions are met, so the series converges by the Alternating Series Test.
Consider the series:
\[ \sum_{n=1}^{\infty} (-1)^{n-1} \frac{\ln(n)}{n} = \frac{\ln(1)}{1} – \frac{\ln(2)}{2} + \frac{\ln(3)}{3} – \frac{\ln(4)}{4} + \dots \]
Here, \(a_n = \frac{\ln(n)}{n}\).
\[ \frac{\ln(n+1)}{n+1} \leq \frac{\ln(n)}{n} \]
This can be confirmed through calculus or inequalities, but it is not immediately obvious.
\[ \lim_{n \to \infty} \frac{\ln(n)}{n} = 0 \]
This requires L’Hôpital’s Rule to prove, as it involves an indeterminate form. However, once shown that both conditions are satisfied, the series converges by the Alternating Series Test.
The Alternating Series Test is a powerful method for determining the convergence of series that alternate in sign. By applying this test, you can confidently determine whether a series converges by checking two key conditions: decreasing terms and the limit approaching zero. The examples provided range from simple to complex, illustrating the versatility of this test in various contexts.