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Magazine Chapter 4: Problem 7
Determine whether \(\sum_{n \geq 2} a_{n}\) converges, when \(a_{n}\) is given as below. 1\. \(\frac{(-1)^{n}}{\sqrt{n^{2}+n}}\) 4\. \(\frac{(-1)^{n}}{\log n+\sin (2 n \pi / 3)}\) 2\. \(\frac{(-1)^{n}}{\log n}\) 5\. \(\sqrt{1+\frac{(-1)^{n}}{\sqrt{n}}}-1\) 3\. \(\frac{1+(-1)^{n} \sqrt{n}}{n}\) 6\. \(\frac{(-1)^{n}}{\sqrt{n}+(-1)^{n}}\)
Short Answer
Expert verified
Series 1, 2, 4, 5, 6 converge. Series 3 does not.
Step by step solution
01
Examine Series 1
We have the series \(a_n = \frac{(-1)^n}{\sqrt{n^2 + n}}\). Since \(n^2 + n \approx n^2\) for large \(n\), we can approximate \(a_n \approx \frac{(-1)^n}{n}\). This form suggests an alternating series with terms approximately \(\frac{1}{n}\), similar to the alternating harmonic series, which converges.
02
Examine Series 2
Consider the series \(a_n = \frac{(-1)^n}{\log n}\). For this alternating series to converge by the alternating series test, \(|a_n|\) should decrease monotonically to zero. However, as \(n \to \infty\), \(\log n\) increases. Hence, \(\frac{1}{\log n} \to 0\) and is decreasing. Therefore, the series converges.
03
Examine Series 3
The series is \(a_n = \frac{1 + (-1)^n \sqrt{n}}{n}\). For even \(n\), \(a_{2n} = \frac{1 + \sqrt{n}}{n}\) and for odd \(n\), \(a_{2n+1} = \frac{1 - \sqrt{n}}{n}\). Since the terms do not tend to zero, the series does not converge.
04
Examine Series 4
Inspect the series \(a_n = \frac{(-1)^n}{\log n + \sin(2n\pi/3)}\). For large \(n\), the denominator oscillates based on the \(\sin\) term, but \(\log n\) ensures it's bounded away from zero. The conditions for the alternating series test are met, ensuring convergence.
05
Examine Series 5
For \(a_n = \sqrt{1 + \frac{(-1)^n}{\sqrt{n}}} - 1\), use the binomial expansion: \(\sqrt{1 + x} \approx 1 + \frac{x}{2}\) for small \(x\), implying \(\sqrt{1 + \frac{(-1)^n}{\sqrt{n}}} \approx 1 + \frac{(-1)^n}{2\sqrt{n}}\). Therefore, \(a_n \approx \frac{(-1)^n}{2\sqrt{n}}\), an alternating convergent series.
06
Examine Series 6
For \(a_n = \frac{(-1)^n}{\sqrt{n} + (-1)^n}\), notice \(\sqrt{n} + (-1)^n \approx \sqrt{n}\) for large \(n\). Thus, \(a_n \approx \frac{(-1)^n}{\sqrt{n}}\), an alternating series that converges.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convergence
Convergence is an essential concept in calculus and mathematical analysis, referring to the behavior of a series as the number of terms increases indefinitely. When we say a series converges, it means that as you add more and more terms, the series approaches a specific value. In other words, it settles down to a number rather than heading off to infinity or oscillating indefinitely. For instance:
- If a series' terms get smaller rapidly enough as you add them, the series is likely to converge.
- For a series like the harmonic series \( \sum \frac{1}{n} \), it does not converge, whereas the alternating harmonic series converges due to its alternating nature.
- Converging series have a finite sum, while diverging ones do not.
Alternating Series Test
The alternating series test is a handy tool for determining the convergence of series that alternate in sign. This test asserts that an alternating series converges if two conditions are met:
- The absolute value of the terms decreases monotonically (it gets smaller with each term).
- The limit of the absolute terms goes to zero as they approach infinity.
Binomial Expansion
Binomial expansion allows us to tackle expressions raised to a power and is often used to approximate terms for convergence checks. With small values of the variable, binomial expansion can simplify the form of expressions such as:
- For expressions resembling \( \sqrt{1 + x} \), when \( x \) is small, use the expansion \( \approx 1 + \frac{x}{2} - \frac{x^2}{8} + \ldots \).
- This helps in approximating and simplifying series, making it easier to analyze their convergence properties.
Harmonic Series
The harmonic series, denoted as \( \sum \frac{1}{n} \), serves as a foundational example of a divergent series. It is made up of the reciprocals of natural numbers:
- This series grows without bound, implying it diverges rather than converging to a finite sum.
- Understanding the harmonic series is crucial when analyzing other series that are similar in form. For example, alterations like alternating signs can sometimes turn a divergent series into a convergent one, as seen in the alternating harmonic series.
- In application, the harmonic series shows up in many areas of science and engineering, often serving as a caution for ensuring conditions are met for convergence.
