Problem 30
Question
In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \sin \frac{\pi}{n} $$
Step-by-Step Solution
Verified Answer
The series is conditionally convergent.
1Step 1: Determine Type of Series
The series given is \( \sum_{n=1}^{\infty}(-1)^{n+1} \sin \frac{\pi}{n} \). This is an alternating series because it has a \((-1)^{n+1}\) factor, which changes the sign of the terms between positive and negative as \(n\) increases.
2Step 2: Check for Absolute Convergence
To check for absolute convergence, consider the series without the alternating factor: \( \sum_{n=1}^{\infty} \left| \sin \frac{\pi}{n} \right| \). As \( n \to \infty \), \( \sin \frac{\pi}{n} \approx \frac{\pi}{n} \). Therefore, this resembles the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is known to diverge. Hence, the series does not converge absolutely.
3Step 3: Apply Alternating Series Test for Conditional Convergence
The alternating series test requires two conditions: \( a_n \) (where \( a_n = \sin \frac{\pi}{n} \)) is decreasing and approaches zero as \( n \to \infty \). Since \( \sin \frac{\pi}{n} \approx \frac{\pi}{n} \) and decreases as \( n \) increases, and \( \lim_{n \to \infty} \sin \frac{\pi}{n} = 0 \), both conditions are satisfied. Thus, the series is conditionally convergent.
Key Concepts
Absolute ConvergenceConditional ConvergenceHarmonic SeriesAlternating Series Test
Absolute Convergence
Absolute convergence refers to a situation where the absolute values of the terms of a series sum up to a convergent series. Let's break this down a bit more. When we have a series like \( \sum_{n=1}^{\infty} a_n \), we check the convergence of \( \sum_{n=1}^{\infty} |a_n| \) to determine absolute convergence. If this series converges, the original series \( \sum_{n=1}^{\infty} a_n \) is said to absolutely converge.
For the series in our original exercise, \( \sum_{n=1}^{\infty} (-1)^{n+1} \sin \frac{\pi}{n} \), we looked at \( \sum_{n=1}^{\infty} |\sin \frac{\pi}{n}| \). This resembles a very famous series, the harmonic series, as \( \sin \frac{\pi}{n} \approx \frac{\pi}{n} \) for large \( n \).
The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges. Thus, the series \( \sum_{n=1}^{\infty} |\sin \frac{\pi}{n}| \) also diverges. Since it doesn't converge absolutely, \( \sum_{n=1}^{\infty} (-1)^{n+1} \sin \frac{\pi}{n} \) is not absolutely convergent.
For the series in our original exercise, \( \sum_{n=1}^{\infty} (-1)^{n+1} \sin \frac{\pi}{n} \), we looked at \( \sum_{n=1}^{\infty} |\sin \frac{\pi}{n}| \). This resembles a very famous series, the harmonic series, as \( \sin \frac{\pi}{n} \approx \frac{\pi}{n} \) for large \( n \).
The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges. Thus, the series \( \sum_{n=1}^{\infty} |\sin \frac{\pi}{n}| \) also diverges. Since it doesn't converge absolutely, \( \sum_{n=1}^{\infty} (-1)^{n+1} \sin \frac{\pi}{n} \) is not absolutely convergent.
Conditional Convergence
When a series is not absolutely convergent, we check for conditional convergence. Conditional convergence occurs when a series converges, but does not meet the criteria for absolute convergence.
In our case, the alternating series test comes in handy here. Though \( \sum_{n=1}^{\infty} |\sin \frac{\pi}{n}| \) diverges, we need to see if the alternating series itself converges.
In this exercise, after stripping away the absolute values, the series becomes \( \sum_{n=1}^{\infty} (-1)^{n+1} \sin \frac{\pi}{n} \). The conditions for conditional convergence using the alternating series test are:
In our case, the alternating series test comes in handy here. Though \( \sum_{n=1}^{\infty} |\sin \frac{\pi}{n}| \) diverges, we need to see if the alternating series itself converges.
In this exercise, after stripping away the absolute values, the series becomes \( \sum_{n=1}^{\infty} (-1)^{n+1} \sin \frac{\pi}{n} \). The conditions for conditional convergence using the alternating series test are:
- The function \( a_n = \sin \frac{\pi}{n} \) must decrease; which it does since \( \sin \frac{\pi}{n} \approx \frac{\pi}{n} \) as \( n \) becomes large.
- \( \lim_{n \to \infty} \sin \frac{\pi}{n} = 0 \), which is also true.
Harmonic Series
The harmonic series is a fascinating and fundamental concept in calculus. Defined as \( \sum_{n=1}^{\infty} \frac{1}{n} \), it is infamous for diverging. Even though the terms \( \frac{1}{n} \) get very small as \( n \) increases, they never add up to a fixed number.
This is crucial because the harmonic series helps us understand absolute convergence. When a series is compared to it, like \( \sin \frac{\pi}{n} \approx \frac{\pi}{n} \) in our exercise, it informs us about divergence.
Understanding this behavior is key to identifying whether a series behaves like a harmonic one and thus diverges, informing our decision on absolute convergence.
This is crucial because the harmonic series helps us understand absolute convergence. When a series is compared to it, like \( \sin \frac{\pi}{n} \approx \frac{\pi}{n} \) in our exercise, it informs us about divergence.
Understanding this behavior is key to identifying whether a series behaves like a harmonic one and thus diverges, informing our decision on absolute convergence.
Alternating Series Test
The Alternating Series Test is an essential tool in determining the convergence of alternating series. Alternating series are those in which the terms alternatively add and subtract, like our original series \( \sum_{n=1}^{\infty} (-1)^{n+1} \sin \frac{\pi}{n} \).
To apply this test, we ensure two conditions are met:
Both conditions are fulfilled in our case, proving conditional convergence. The test is highly effective because it allows some series to converge even when they would diverge if the terms were not alternating. It's a nuanced way to see how the interplay of sequential positive and negative terms can lead to convergence.
To apply this test, we ensure two conditions are met:
- The terms of the series \( a_n \), here \( \sin \frac{\pi}{n} \), must be decreasing in absolute value.
- The limit \( \lim_{n \to \infty} a_n = 0 \).
Both conditions are fulfilled in our case, proving conditional convergence. The test is highly effective because it allows some series to converge even when they would diverge if the terms were not alternating. It's a nuanced way to see how the interplay of sequential positive and negative terms can lead to convergence.
Other exercises in this chapter
Problem 30
Calculate, accurate to five decimal places, $$ \int_{0}^{0.5} \sin \sqrt{x} d x $$
View solution Problem 30
\(\sum_{n=1}^{\infty} \frac{5^{2 n}}{n !}\)
View solution Problem 30
Suppose that \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}=\sum_{n=0}^{\infty} b_{n} x^{n}\) for \(|x|
View solution Problem 31
Find the radius of convergence of $$ \sum_{n=1}^{\infty} \frac{1 \cdot 2 \cdot 3 \cdots n}{1 \cdot 3 \cdot 5 \cdots(2 n-1)} x^{2 n+1} $$
View solution