Problem 27
Question
Determine convergence or divergence for each of the series. Indicate the test you use. \(\sum_{n=1}^{\infty} \frac{1}{2+\sin ^{2} n}\)
Step-by-Step Solution
Verified Answer
The series \( \sum_{n=1}^{\infty} \frac{1}{2+\sin^{2} n} \) diverges by the comparison test.
1Step 1: Identify the Type of Series
The given series is \( \sum_{n=1}^{\infty} \frac{1}{2+\sin^{2}n} \). It is an infinite series with the general term \( a_n = \frac{1}{2+\sin^{2}n} \).
2Step 2: Analyze the General Term
The general term \( a_n = \frac{1}{2+\sin^2n} \) is positive for all \( n \), since \( \sin^2n \geq 0 \) and therefore \( 2 + \sin^2n \geq 2 \). This makes \( a_n \) non-negative.
3Step 3: Decide a Convergence Test
Observe that \( 0 < \sin^2n \leq 1 \), so \( 2 \leq 2+\sin^2n \leq 3 \). This results in \( \frac{1}{3} \leq \frac{1}{2+\sin^2n} \leq \frac{1}{2} \), suggesting the series shares properties with a inndeterminate harmonic series.
4Step 4: Apply the Limit Comparison Test
Let's compare \( a_n = \frac{1}{2+\sin^2n} \) with the harmonic series \( b_n = \frac{1}{n} \). Note that for large \( n \), \( \frac{1}{2} \approx a_n \) bounds the terms from below.
5Step 5: Determine Result Using the Comparison Test
Since \( a_n \approx \frac{1}{2} > 0 \) does not converge to 0 (since \( \sum \frac{1}{n} \) diverges, \( \sum a_n \) must also not converge), the direct comparison test leads to divergence.
Key Concepts
Limit Comparison TestInfinite SeriesHarmonic Series
Limit Comparison Test
The Limit Comparison Test is a handy tool to determine if an infinite series converges or diverges by comparing it to a second, more familiar series. This test typically comes into play when working with series whose terms are non-negative and often involves series like the
You compare the given series, say \( \sum a_n \), to another series \( \sum b_n \) where the behavior is known. Calculate the limit of the ratio \( \lim_{{n \to \infty}} \frac{a_n}{b_n} \). If this limit \( L \) is a finite number and positive \( (0 < L < \infty) \), then \( \sum a_n \) and \( \sum b_n \) either both converge or both diverge.
In our exercise, the given series \( \sum \frac{1}{2 + \sin^2n} \) was compared to the harmonic series \( \sum \frac{1}{n} \). Since the harmonic series diverges and the limit comparison shows this series behaves similarly, the original series diverges too.
- Geometric series
- p-series
- Harmonic series
You compare the given series, say \( \sum a_n \), to another series \( \sum b_n \) where the behavior is known. Calculate the limit of the ratio \( \lim_{{n \to \infty}} \frac{a_n}{b_n} \). If this limit \( L \) is a finite number and positive \( (0 < L < \infty) \), then \( \sum a_n \) and \( \sum b_n \) either both converge or both diverge.
In our exercise, the given series \( \sum \frac{1}{2 + \sin^2n} \) was compared to the harmonic series \( \sum \frac{1}{n} \). Since the harmonic series diverges and the limit comparison shows this series behaves similarly, the original series diverges too.
Infinite Series
An infinite series is an expression involving an infinite sum of terms. Each infinite series has the form \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the nth term of the series.
The convergence or divergence of an infinite series often relies on the behavior of its terms as \( n \) approaches infinity.
For practical purposes, understanding the behavior of common series types like geometric or harmonic series provides useful insight when evaluating less typical series.
The convergence or divergence of an infinite series often relies on the behavior of its terms as \( n \) approaches infinity.
- If the sum approaches a specific value, the series is said to converge.
- If the sum tends to grow indefinitely, or oscillates without settling, the series diverges.
For practical purposes, understanding the behavior of common series types like geometric or harmonic series provides useful insight when evaluating less typical series.
Harmonic Series
The harmonic series is a well-known example of a divergent infinite series, expressed as \( \sum_{n=1}^{\infty} \frac{1}{n} \).
Despite its simple form, it has a curious property of divergence, meaning as more terms are added, the sum keeps increasing indefinitely without bounds. This makes it a useful benchmark in convergence tests.
Key characteristics of the harmonic series include:
Despite its simple form, it has a curious property of divergence, meaning as more terms are added, the sum keeps increasing indefinitely without bounds. This makes it a useful benchmark in convergence tests.
Key characteristics of the harmonic series include:
- The series grows without bound as more terms are included.
- Each added term is positive and decreases in size (as \( n \) increases, \( \frac{1}{n} \) becomes smaller).
- Though the terms shrink, they don't approach zero fast enough to allow for convergence.
Other exercises in this chapter
Problem 26
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