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Magazine Chapter 10: Problem 17
Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{1}{\sqrt{n}}$$
Short Answer
Expert verified
The series converges conditionally.
Step by step solution
01
Identify the Type of Series
The given series is \( \sum_{n=1}^{\infty} (-1)^n \frac{1}{\sqrt{n}} \). This is an alternating series because of the term \((-1)^n\).
02
Determine Absolute Convergence
To check for absolute convergence, consider the series of absolute values: \( \sum_{n=1}^{\infty} \left| (-1)^n \frac{1}{\sqrt{n}} \right| = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \). This series is a p-series with \( p = \frac{1}{2} \), which is known to diverge because \( p \leq 1 \).
03
Apply the Alternating Series Test
Since the series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) diverges, check if the original series meets the conditions of the Alternating Series Test. The test requires that the terms \( \frac{1}{\sqrt{n}} \) are decreasing and approach zero as \( n \to \infty \).
04
Check Conditions of the Alternating Series Test
The terms \( \frac{1}{\sqrt{n}} \) do indeed decrease and approach zero as \( n \to \infty \). Therefore, the alternating series test conditions are satisfied.
05
Conclude Final Result
Since the alternating series test is satisfied but the series does not converge absolutely, the given series converges conditionally.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Alternating Series
Alternating series are series where the signs of the terms switch from positive to negative as the series progresses. This is due to the presence of a factor like \((-1)^n\) in the series.
In our example, it is expressed as \[\sum_{n=1}^{\infty} (-1)^n \frac{1}{\sqrt{n}}\].
The alternating nature involves:
In our example, it is expressed as \[\sum_{n=1}^{\infty} (-1)^n \frac{1}{\sqrt{n}}\].
The alternating nature involves:
- Having terms alternate in their sign (+/-).
- A common term to identify such series is \((-1)^n\) or \((-1)^{n+1}\), which switches from positive to negative (or vice versa) as \ n \ changes.
Absolute Convergence
A series is said to converge absolutely if the series of its absolute terms converges. To test this, simply take the absolute value of each term in the series and determine if this new series converges.
For example, consider our given series \[\sum_{n=1}^{\infty} (-1)^n \frac{1}{\sqrt{n}}\].
To check for absolute convergence, we focus on \[\sum_{n=1}^{\infty} \left| (-1)^n \frac{1}{\sqrt{n}} \right| = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\].
For example, consider our given series \[\sum_{n=1}^{\infty} (-1)^n \frac{1}{\sqrt{n}}\].
To check for absolute convergence, we focus on \[\sum_{n=1}^{\infty} \left| (-1)^n \frac{1}{\sqrt{n}} \right| = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\].
- If this series converges, then our original series converges absolutely.
- If it diverges, the original series may still converge, but only conditionally.
p-series
A p-series is a type of series that takes the form \[\sum_{n=1}^{\infty} \frac{1}{n^p}\].
Here, the value of \ p \ determines the behavior of the series:
Since \ p = \frac{1}{2} \, which is less than or equal to 1, this series does not converge, indicating divergence.
This understanding helps in deciding whether an alternating series can converge absolutely by checking its p-series nature.
Here, the value of \ p \ determines the behavior of the series:
- If \ p > 1 \, the series converges.
- If \ p \leq 1 \, the series diverges.
Since \ p = \frac{1}{2} \, which is less than or equal to 1, this series does not converge, indicating divergence.
This understanding helps in deciding whether an alternating series can converge absolutely by checking its p-series nature.
Divergence
Divergence occurs when a series does not have a finite sum, meaning it fails to approach a single value as more and more terms are added.
For our series \[\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\],
This concept is vital when determining whether a series converges or if we're simply observing its endless growth.
For our series \[\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\],
- Since it is a p-series with \ p = \frac{1}{2} \, it diverges.
- A divergent series means its partial sums grow indefinitely as \ n \ increases.
This concept is vital when determining whether a series converges or if we're simply observing its endless growth.
Alternating Series Test
The Alternating Series Test helps determine conditional convergence for those series that alternate in signs. To apply this test, two main conditions must be satisfied:
- This indeed decreases over time.- It approaches zero as \( n \) grows large.
Thus, the conditions are met. Therefore, despite the series not absolutely converging, it converges conditionally thanks to the Alternating Series Test.
This test is invaluable for handling series with fluctuating signs.
- The absolute value of the terms \( b_n \) should decrease progressively.\[b_{n} \geq b_{n+1}\]
- The limit of \( b_n \) as \( n \) approaches infinity should equal zero. \[\lim_{{n \to \infty}} b_n = 0\]
- This indeed decreases over time.- It approaches zero as \( n \) grows large.
Thus, the conditions are met. Therefore, despite the series not absolutely converging, it converges conditionally thanks to the Alternating Series Test.
This test is invaluable for handling series with fluctuating signs.
