Problem 37
Question
Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \ln \left( \frac {n^2 + 1}{2n^2 + 1} \right) \)
Step-by-Step Solution
Verified Answer
The series is divergent.
1Step 1: Analyze the Terms
Let's analyze the general term of the series: \( a_n = \ln \left( \frac{n^2 + 1}{2n^2 + 1} \right) \). As \( n \) approaches infinity, \( a_n \) seems to approach \( \ln \left( \frac{1}{2} \right) \).
2Step 2: Apply the Divergence Test
According to the Divergence Test, if \( \lim_{{n \to \infty}} a_n eq 0 \), then the series \( \sum_{n=1}^{\infty} a_n \) diverges. Calculate this limit: \( \lim_{{n \to \infty}} \ln \left( \frac{n^2 + 1}{2n^2 + 1} \right) = \ln \left( \frac{1}{2} \right) eq 0 \). Thus, the series diverges.
Key Concepts
Divergence TestInfinite SeriesLimit of a Sequence
Divergence Test
The divergence test is a straightforward tool used in determining the behavior of infinite series. An infinite series is an extensive sum that adds up each term in a sequence, extending indefinitely. The divergence test hinges on evaluating the limit of the sequence of terms that compose the series.
To determine whether a series diverges, the divergence test examines the limit of its terms as the series progresses towards infinity. Specifically, for a series given by \( \sum_{n=1}^{\infty} a_n \), if \( \lim_{{n \to \infty}} a_n eq 0 \), the series diverges. The zero-limit condition must be met for further tests to be required for convergence.
In simpler terms, if the individual terms of a series don't shrink to zero as \( n \) approaches infinity, the entire series cannot settle into a finite sum. For the given exercise, since the limit of \( a_n = \ln \left( \frac{n^2 + 1}{2n^2 + 1} \right) \) approaches \( \ln \left( \frac{1}{2} \right) \), which isn't zero, it signals unequivocally that the series diverges.
To determine whether a series diverges, the divergence test examines the limit of its terms as the series progresses towards infinity. Specifically, for a series given by \( \sum_{n=1}^{\infty} a_n \), if \( \lim_{{n \to \infty}} a_n eq 0 \), the series diverges. The zero-limit condition must be met for further tests to be required for convergence.
In simpler terms, if the individual terms of a series don't shrink to zero as \( n \) approaches infinity, the entire series cannot settle into a finite sum. For the given exercise, since the limit of \( a_n = \ln \left( \frac{n^2 + 1}{2n^2 + 1} \right) \) approaches \( \ln \left( \frac{1}{2} \right) \), which isn't zero, it signals unequivocally that the series diverges.
Infinite Series
Infinite series are sums of infinitely many terms. They are written in the form \( \sum_{n=1}^{\infty} a_n \), where the sequence \( a_n \) denotes the terms of the series. Understanding an infinite series involves exploring whether these sums lead to a singular value (convergence) or spiral away without limit (divergence).
Infinite series can either converge or diverge based on the behavior of their terms. When a series converges, it can be summed to a specific number; this is known as the sum of the series. In contrast, a divergent series fails to approach any finite limit and therefore does not sum to any specific value.
Examples of well-known infinite series include geometric series and harmonic series, each possessing distinct properties regarding convergence. The convergence of a series doesn't depend solely upon having infinitely many terms, but rather on the nature and behavior of those terms as they accumulate.
Infinite series can either converge or diverge based on the behavior of their terms. When a series converges, it can be summed to a specific number; this is known as the sum of the series. In contrast, a divergent series fails to approach any finite limit and therefore does not sum to any specific value.
Examples of well-known infinite series include geometric series and harmonic series, each possessing distinct properties regarding convergence. The convergence of a series doesn't depend solely upon having infinitely many terms, but rather on the nature and behavior of those terms as they accumulate.
Limit of a Sequence
To fully grasp concepts related to series and convergence, understanding the limit of a sequence is fundamental. A sequence is a list of numbers in a specific order, and its limit describes the number that the terms of the sequence approach as the index increases.
Formally, the limit of a sequence \( \{a_n\} \) is \( L \) if for every positive number \( \epsilon \), there exists a value \( N \) such that for all \( n > N \), \( |a_n - L| < \epsilon \). This notion captures the idea that terms get arbitrarily close to \( L \) as the sequence progresses.
In analyzing series, discerning the limit of individual terms helps determine the convergence or divergence of the entire series. If the sequence of terms \( a_n \) does not tend towards zero, the series \( \sum a_n \) can't converge to a finite value. Thus, understanding limits is critical in applying the divergence test and other approaches that evaluate series behavior.
Formally, the limit of a sequence \( \{a_n\} \) is \( L \) if for every positive number \( \epsilon \), there exists a value \( N \) such that for all \( n > N \), \( |a_n - L| < \epsilon \). This notion captures the idea that terms get arbitrarily close to \( L \) as the sequence progresses.
In analyzing series, discerning the limit of individual terms helps determine the convergence or divergence of the entire series. If the sequence of terms \( a_n \) does not tend towards zero, the series \( \sum a_n \) can't converge to a finite value. Thus, understanding limits is critical in applying the divergence test and other approaches that evaluate series behavior.
Other exercises in this chapter
Problem 37
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