Alternating series test

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

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:

1. Decreasing Terms: The sequence \(a_n\) is decreasing, i.e., \(a_{n+1} \leq a_n\) for all \(n\).
2. Limit of Terms Approaching Zero: The limit of \(a_n\) as \(n\) approaches infinity is zero:

\[ \lim_{n \to \infty} a_n = 0 \]

If both conditions hold, the series converges.

Example 1: Easy Level

Consider the series:

\[ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} = 1 – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + \dots \]

Here, \(a_n = \frac{1}{n}\).

• Decreasing Terms: The sequence \(\frac{1}{n}\) is decreasing since \(\frac{1}{n+1} \leq \frac{1}{n}\).
• Limit of Terms Approaching Zero: The limit is:

\[ \lim_{n \to \infty} \frac{1}{n} = 0 \]

Since both conditions are satisfied, the series converges by the Alternating Series Test. This is known as the alternating harmonic series.

Example 2: Medium Level

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, \(a_n = \frac{1}{n^2}\).

• Decreasing Terms: The sequence \(\frac{1}{n^2}\) is decreasing since \(\frac{1}{(n+1)^2} \leq \frac{1}{n^2}\).
• Limit of Terms Approaching Zero: The limit is:

\[ \lim_{n \to \infty} \frac{1}{n^2} = 0 \]

Again, both conditions are met, so the series converges by the Alternating Series Test.

Example 3: Hard Level

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, \(a_n = \frac{\ln(n)}{n}\).

• Decreasing Terms: We need to check if \(a_n\) is decreasing. The sequence \(\frac{\ln(n)}{n}\) decreases if:

\[ \frac{\ln(n+1)}{n+1} \leq \frac{\ln(n)}{n} \]

This can be confirmed through calculus or inequalities, but it is not immediately obvious.

• Limit of Terms Approaching Zero: The limit is:

\[ \lim_{n \to \infty} \frac{\ln(n)}{n} = 0 \]

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.

Conclusion

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.