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Magazine Chapter 11: Problem 17
Determine whether the alternating series in converge or diverge. \(\sum_{n=1}^{*} \frac{(-1)^{n+1}}{\sqrt[n]{2}}\)
Short Answer
Expert verified
The series converges by the alternating series test.
Step by step solution
01
Apply the Alternating Series Test Conditions
For the series \(\sum (-1)^{n+1} a_n\), two conditions must be met to determine convergence using the alternating series test: (1) The terms \(a_n\) must be positive, decreasing, and approach zero as \(n\to\infty\). Here, \(a_n = \frac{1}{\sqrt[n]{2}} \). This is positive for all \(n > 0\).
02
Check if \(a_n\) is Decreasing
To check if \(a_n = \frac{1}{\sqrt[n]{2}}\) is decreasing, we need to check if \(a_{n+1} < a_n\) for all \(n\). That is, \(\frac{1}{\sqrt[n+1]{2}} < \frac{1}{\sqrt[n]{2}}\). By simplifying, \(\sqrt{\frac{\sqrt[n+1]{2}}{\sqrt[n]{2}}}< 1\), this implies \(\sqrt[n+1]{2} > \sqrt[n]{2}\), which verifiable. Thus, \(a_n\) is decreasing.
03
Verify \(a_n\) Approaches Zero
Finally, check if \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{\sqrt[n]{2}}=0\). As \(n\) increases to infinity, \(\sqrt[n]{2}\) becomes larger, thus \(\frac{1}{\sqrt[n]{2}}\) approaches zero.
04
Conclude the Convergence
Since the conditions for the alternating series test are satisfied鈥擻(a_n\) is positive, decreasing, and approaches zero鈥攖he series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt[n]{2}}\) converges.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Series Convergence
In mathematics, understanding when a series converges is crucial to many areas of study. A series converges when its terms approach a single value as the number of terms goes to infinity. This means that the sum of the series does not grow indefinitely but settles towards a certain number. To determine if a series converges, several tests and criteria can be applied, one of them being the Alternating Series Test.
A convergent series has significant implications:
A convergent series has significant implications:
- It implies stability and boundedness of the sum.
- Ensures the series has a "limiting sum," which can be interpreted numerically.
Decreasing Sequence
A sequence is referred to as decreasing if each term is smaller than the previous term. Mathematically, a sequence \( \{a_n\} \) is decreasing if \( a_{n+1} < a_n \) for all \( n \). For the Alternating Series Test to indicate convergence, the sequence \( a_n \) must be decreasing.
In this exercise, we see how \( a_n = \frac{1}{\sqrt[n]{2}} \) satisfies this condition. Here is how you can check that it is decreasing:
In this exercise, we see how \( a_n = \frac{1}{\sqrt[n]{2}} \) satisfies this condition. Here is how you can check that it is decreasing:
- Evaluate the inequality \( \sqrt[n+1]{2} > \sqrt[n]{2} \).
- This simplification ensures that subsequent terms of \( a_n \) are smaller.
Limit of a Sequence
The limit of a sequence is crucial in understanding the behavior of a series as its terms progress towards infinity. For a sequence \( \{a_n\} \) with limit \( L \), \( \lim_{n \to \infty} a_n = L \). If the limit is zero, the sequence diminishes over time, which is a requirement for the convergence of alternating series.
When we analyze \( a_n = \frac{1}{\sqrt[n]{2}} \), calculating its limit involves observing how the expression behaves as \( n \) increases:
When we analyze \( a_n = \frac{1}{\sqrt[n]{2}} \), calculating its limit involves observing how the expression behaves as \( n \) increases:
- With increased \( n \), \( \sqrt[n]{2} \) grows larger.
- This causes \( \frac{1}{\sqrt[n]{2}} \) to approach zero.
Alternating Series
Alternating series are characterized by a sequence whose terms switch signs regularly. The series looks like this: \( \sum (-1)^n a_n \). The Alternating Series Test provides a useful way to test for convergence with alternating series. When applied, the test asserts that if the positive sequence \( a_n \) is both decreasing and approaches zero, the alternating series will converge.
Let's view the key aspects for applying the test successfully:
Let's view the key aspects for applying the test successfully:
- Identify the positive part of the term, \( a_n \).
- Ensure that \( a_n \) is decreasing.
- Check that \( \lim_{n \to \infty} a_n = 0 \).
