Problem 3
Question
Let $$ a_{n}=\frac{2 n}{10 n+7} $$ For the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter 'infinity' if it diverges to \(\infty\), '-infinity' if it diverges to \(-\infty\) or 'DNE' otherwise. (a) The series \(\sum_{n=1}^{\infty} \frac{2 n}{10 n+7}\). (b) The sequence \(\left\\{\frac{2 n}{10 n+7}\right\\}\).
Step-by-Step Solution
Verified Answer
(a) Diverges (DNE)(b) Converges (\( \frac{1}{5} \))
1Step 1: Analyze the Sequence
First, consider the sequence given by \(a_{n} = \frac{2n}{10n+7}\). To determine its behavior as \( n \to \infty \), analyze the expression.
2Step 2: Simplify the Expression
Divide numerator and denominator by \( n \):\[ a_{n} = \frac{2n}{10n+7} = \frac{2}{10 + \frac{7}{n}} \]
3Step 3: Determine the Limit of the Sequence
As \( n \to \infty \), the term \( \frac{7}{n} \) approaches 0. Hence, the sequence simplifies to:\[ a_{n} = \frac{2}{10} = \frac{1}{5} \]Therefore, the sequence converges to \( \frac{1}{5} \).
4Step 4: Analyze the Series
Consider the series \( \sum_{n=1}^{\infty} \frac{2n}{10n+7} \). For large \( n \), the general term \( \frac{2n}{10n+7} \approx \frac{1}{5} \).
5Step 5: Apply the Divergence Test
For a series \( \sum_{n=1}^{\infty} a_{n} \) to converge, the general term \( a_{n} \) must approach 0 as \( n \to \infty \). Here, \( \frac{2n}{10n+7} \) approaches \( \frac{1}{5} \), which is not 0. Therefore, the series diverges.
Key Concepts
ConvergenceDivergenceLimit of a SequenceDivergence Test
Convergence
Convergence refers to the behavior of a sequence or series as the number of terms goes towards infinity. A sequence is said to converge if its terms approach a specific value as the index increases. In this context, for a sequence \(a_n = \frac{2n}{10n+7}\), we analyze its limit as \(n\) approaches infinity.
By simplifying the expression, we find that the sequence tends towards \(\frac{1}{5}\). Therefore, the sequence \(\left\{\frac{2n}{10n+7}\right\}\) converges to \(\frac{1}{5}\). Understanding convergence helps us predict the long-term behavior of sequences and series, which is crucial in various fields such as calculus and mathematical analysis.
By simplifying the expression, we find that the sequence tends towards \(\frac{1}{5}\). Therefore, the sequence \(\left\{\frac{2n}{10n+7}\right\}\) converges to \(\frac{1}{5}\). Understanding convergence helps us predict the long-term behavior of sequences and series, which is crucial in various fields such as calculus and mathematical analysis.
Divergence
Divergence occurs when a sequence or series does not approach a specific limit as the number of terms increases. In our exercise, we consider the series \(\sum_{n=1}^{\infty} \frac{2n}{10n+7}\).
For a series to converge, its terms must approach zero as \(n\) increases. In our case, the terms of the series approach \(\frac{1}{5}\), which is not zero. Thus, the series diverges. Recognizing divergence is important because it indicates that the sum or the limit does not exist in the finite sense, which can greatly affect calculations and interpretations in higher mathematics.
For a series to converge, its terms must approach zero as \(n\) increases. In our case, the terms of the series approach \(\frac{1}{5}\), which is not zero. Thus, the series diverges. Recognizing divergence is important because it indicates that the sum or the limit does not exist in the finite sense, which can greatly affect calculations and interpretations in higher mathematics.
Limit of a Sequence
The limit of a sequence describes the value that the terms of the sequence approach as the index becomes very large. For the sequence \(a_n = \frac{2n}{10n+7}\), we simplify the expression to better understand its limit:
By dividing both the numerator and the denominator by \(n\), we find that \(a_n = \frac{2}{10 + \frac{7}{n}}\). As \(n\) approaches infinity, the term \(\frac{7}{n}\) approaches zero. This simplifies the expression to \(a_n = \frac{2}{10} = \frac{1}{5}\).
Therefore, the sequence converges to \(\frac{1}{5}\). Understanding the limit of a sequence is fundamental in calculus as it directly relates to continuity, differentiability, and integrals.
By dividing both the numerator and the denominator by \(n\), we find that \(a_n = \frac{2}{10 + \frac{7}{n}}\). As \(n\) approaches infinity, the term \(\frac{7}{n}\) approaches zero. This simplifies the expression to \(a_n = \frac{2}{10} = \frac{1}{5}\).
Therefore, the sequence converges to \(\frac{1}{5}\). Understanding the limit of a sequence is fundamental in calculus as it directly relates to continuity, differentiability, and integrals.
Divergence Test
The Divergence Test is a simple but powerful tool to determine if a series \((\sum a_n)\) diverges. According to this test, if the limit of \(a_n\) as \(n\) approaches infinity is not zero, then the series must diverge.
In our case, the series \(\sum_{n=1}^{\infty} \frac{2n}{10n+7}\) approaches \(\frac{1}{5}\) as \(n\) increases. Because \( \frac{1}{5} eq 0\), the Divergence Test confirms that this series diverges.
Using the Divergence Test can quickly indicate whether further analysis is necessary or if the series immediately simplifies to a divergent one.
In our case, the series \(\sum_{n=1}^{\infty} \frac{2n}{10n+7}\) approaches \(\frac{1}{5}\) as \(n\) increases. Because \( \frac{1}{5} eq 0\), the Divergence Test confirms that this series diverges.
Using the Divergence Test can quickly indicate whether further analysis is necessary or if the series immediately simplifies to a divergent one.
Other exercises in this chapter
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