Problem 41 - Thomas Calculus

Question

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n}(\sqrt{n+1}-\sqrt{n}) $$

Step-by-Step Solution

Answer

The series converges conditionally, but not absolutely.

Step 1: Analyze Absolute Convergence

To determine absolute convergence, check if \( \sum_{n=1}^{\infty} \left| (-1)^{n} (\sqrt{n+1} - \sqrt{n}) \right| \) converges. Here, \( \left| (-1)^{n} (\sqrt{n+1} - \sqrt{n}) \right| = \sqrt{n+1} - \sqrt{n} \geq 0 \).To approximate \( \sqrt{n+1} - \sqrt{n} \), use the estimate \( \sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n+1} + \sqrt{n}} \).As \( n \to \infty \), the expressions become smaller, so the terms \( \frac{1}{\sqrt{n+1} + \sqrt{n}} \) behave like \( \frac{1}{2\sqrt{n}} \). Hence, \( \sum_{n=1}^{\infty} \left| \sqrt{n+1} - \sqrt{n} \right| \) diverges as it behaves like the divergent \( \sum_{n=1}^{\infty} \frac{1}{n^{1/2}} \). Therefore, the series does not converge absolutely.

Step 2: Check Conditional Convergence with Alternating Series Test

For conditional convergence, we use the Alternating Series Test. Since the original series is \( \sum_{n=1}^{\infty} (-1)^{n} (\sqrt{n+1} - \sqrt{n}) \), this is an alternating series.Check two conditions: 1. \( a_n = (\sqrt{n+1} - \sqrt{n}) \) is positive since \( \sqrt{n+1} > \sqrt{n} \) for all \( n \geq 1 \). 2. \( a_n \to 0 \) as \( n \to \infty \), because \( \sqrt{n+1} - \sqrt{n} \to 0 \).Since these conditions are satisfied, the alternating series \( \sum_{n=1}^{\infty} (-1)^{n} (\sqrt{n+1} - \sqrt{n}) \) converges conditionally.

Step 3: Conclusion

To summarize, the series \( \sum_{n=1}^{\infty}(-1)^{n}(\sqrt{n+1}-\sqrt{n}) \) does not converge absolutely because the absolute series does not converge. However, using the Alternating Series Test, it converges conditionally, as it meets the criteria set by the test.

Key Concepts

Absolute Convergence

When we discuss absolute convergence, we focus on the idea that a series converges absolutely if the series of its absolute values converges. In mathematical terms, if you take a series \( \sum_{n=1}^{\infty} a_n \), it converges absolutely if \( \sum_{n=1}^{\infty} |a_n| \) converges. In the context of our given series, \( \sum_{n=1}^{\infty} (-1)^{n} (\sqrt{n+1} - \sqrt{n}) \), we have to consider \( \sum_{n=1}^{\infty} |(-1)^{n} (\sqrt{n+1} - \sqrt{n})| \), which simplifies to \( \sum_{n=1}^{\infty} (\sqrt{n+1} - \sqrt{n}) \).

To check for absolute convergence, we estimate \( \sqrt{n+1} - \sqrt{n} \) as \( \frac{1}{\sqrt{n+1} + \sqrt{n}} \). As \( n \) becomes very large, this behaves like \( \frac{1}{2\sqrt{n}} \). This is similar to the series \( \sum_{n=1}^{\infty} \frac{1}{n^{1/2}} \), which is known to diverge. As a result, the absolute series diverges. Therefore, the original series does not converge absolutely, since its series of absolute values does not converge.

Conditional Convergence

Conditional convergence is another concept where a series converges, but not absolutely. This means that the series itself converges, but its absolute values do not. For our specific series, we can employ the Alternating Series Test to check for conditional convergence. The series in question, \( \sum_{n=1}^{\infty} (-1)^{n} (\sqrt{n+1} - \sqrt{n}) \), is indeed an alternating series.

The Alternating Series Test requires two conditions to be met:

• The sequence \( a_n = \sqrt{n+1} - \sqrt{n} \) must be positive and decreasing. Clearly, for \( n \geq 1 \), \( \sqrt{n+1} > \sqrt{n} \). As \( n \) increases, \( \sqrt{n+1} - \sqrt{n} \) tends to shrink, satisfying the criteria.
• The limit of \( a_n \) as \( n \) approaches infinity should be zero. Here, as \( n \to \infty \), \( \sqrt{n+1} - \sqrt{n} \) indeed trends towards zero.

Since both conditions hold, the series converges conditionally, implying it converges, but does not do so absolutely.

Series Divergence

Divergence in series occurs when the series does not meet the criteria for convergence, whether absolute or conditional. For the series \( \sum_{n=1}^{\infty} (-1)^{n} (\sqrt{n+1} - \sqrt{n}) \), although it converges conditionally, we found in an earlier analysis that its absolute counterpart diverges.

Therefore, to recap, in the absolute convergence test, because \( \sum_{n=1}^{\infty} (\sqrt{n+1} - \sqrt{n}) \) behaves like \( \sum_{n=1}^{\infty} \frac{1}{n^{1/2}} \), which is a divergent series, the absolute series diverges. Elements such as the series behaving like another well-known divergent series guide us in identifying divergence.

In summary, divergence helps confirm that the original series does not converge absolutely, as the absolute series fails to meet convergence requirements, reaffirming its divergent nature.