Problem 30
Question
Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{2 n+1}{1-3 \sqrt{n}} $$
Step-by-Step Solution
Verified Answer
The sequence diverges as it approaches negative infinity.
1Step 1: Identify the Dominant Terms
To determine the convergence of the sequence \(a_n = \frac{2n+1}{1-3\sqrt{n}}\), first identify the terms that grow the fastest as \(n\) approaches infinity. In the numerator, the term \(2n\) dominates, and in the denominator, the term \(3\sqrt{n}\) dominates.
2Step 2: Approximate the Sequence for Large n
For large \(n\), the sequence can be approximated by \(a_n \approx \frac{2n}{-3\sqrt{n}}\). Simplifying this expression gives \(a_n \approx \frac{2\sqrt{n}}{-3}\).
3Step 3: Analyze the Simplified Expression
The simplified expression \(a_n \approx \frac{2\sqrt{n}}{-3}\) indicates that as \(n\) approaches infinity, \(\sqrt{n}\) also approaches infinity. Therefore, \(a_n\) approaches negative infinity.
4Step 4: Determine Convergence or Divergence
Since the sequence \(a_n\) approaches negative infinity as \(n\) approaches infinity, it diverges. For a sequence to converge, it must approach a finite limit, but \(a_n\) does not.
Key Concepts
Dominant TermsLimit AnalysisDivergence
Dominant Terms
When analyzing sequences, identifying the dominant terms is a crucial step. These terms are the ones that increase or decrease the fastest as the sequence progresses, often determining the behavior of the sequence for large values of n. In the sequence given, \( a_n = \frac{2n+1}{1-3\sqrt{n}} \), we need to identify these dominant components both in the numerator and the denominator.
- **Numerator Analysis**: The numerator \( 2n + 1 \) has the term \( 2n \) as its dominant part. This is because as \( n \) becomes very large, the effect of the constant \( 1 \) becomes insignificant compared to \( 2n \).
- **Denominator Analysis**: In the denominator \( 1 - 3\sqrt{n} \), the term \( -3\sqrt{n} \) grows much faster with large \( n \) compared to the constant \( 1 \). Thus, \( -3\sqrt{n} \) is dominant.
Recognizing dominant terms helps to simplify the sequence for limit analysis, as it provides a clearer view of how the sequence behaves when \( n \) approaches infinity.
- **Numerator Analysis**: The numerator \( 2n + 1 \) has the term \( 2n \) as its dominant part. This is because as \( n \) becomes very large, the effect of the constant \( 1 \) becomes insignificant compared to \( 2n \).
- **Denominator Analysis**: In the denominator \( 1 - 3\sqrt{n} \), the term \( -3\sqrt{n} \) grows much faster with large \( n \) compared to the constant \( 1 \). Thus, \( -3\sqrt{n} \) is dominant.
Recognizing dominant terms helps to simplify the sequence for limit analysis, as it provides a clearer view of how the sequence behaves when \( n \) approaches infinity.
Limit Analysis
After identifying the dominant terms in a sequence, the next step is to perform limit analysis. This involves simplifying the sequence based on the dominant terms to determine its behavior as \( n \) approaches infinity. By focusing on the dominant terms \( 2n \) and \( -3\sqrt{n} \) from our sequence, the expression can be simplified.
Consider the approximation for large \( n \):\[a_n \approx \frac{2n}{-3\sqrt{n}} = \frac{2\sqrt{n}}{-3}\\]
This simplification means we focus solely on the terms that govern the overall shape and direction of the sequence as \( n \to \infty \). The approximation helps highlight that the growth of the sequence is directly tied to \( \sqrt{n} \) multiplied by the constant factor \( \frac{2}{-3} \).
Through limit analysis, it becomes apparent whether the entire sequence approaches a particular point or infinity, thereby assisting in determining convergence or divergence.
Consider the approximation for large \( n \):\[a_n \approx \frac{2n}{-3\sqrt{n}} = \frac{2\sqrt{n}}{-3}\\]
This simplification means we focus solely on the terms that govern the overall shape and direction of the sequence as \( n \to \infty \). The approximation helps highlight that the growth of the sequence is directly tied to \( \sqrt{n} \) multiplied by the constant factor \( \frac{2}{-3} \).
Through limit analysis, it becomes apparent whether the entire sequence approaches a particular point or infinity, thereby assisting in determining convergence or divergence.
Divergence
A sequence is said to diverge if it does not approach a finite limit as \( n \) approaches infinity. In the context of the sequence \( a_n = \frac{2n+1}{1-3\sqrt{n}} \), the simplified form \( \frac{2\sqrt{n}}{-3} \) shows that as \( n \) becomes very large, \( a_n \) moves toward negative infinity.
- **Approaching Infinity**: Here, the component \( \sqrt{n} \) is pivotal. Since \( \sqrt{n} \to \infty \) as \( n \to \infty \), this causes the entire expression to increasingly trend toward negative infinity due to the negative sign from \( -3 \).
- **Lack of a Finite Limit**: For a sequence to converge, it must settle into a specific, finite value. However, since \( a_n \) continues decreasing without bound, there is no such finite limit.
Thus, the sequence diverges because it fails to approach any specific value, illustrating the concept of divergence in sequences.
- **Approaching Infinity**: Here, the component \( \sqrt{n} \) is pivotal. Since \( \sqrt{n} \to \infty \) as \( n \to \infty \), this causes the entire expression to increasingly trend toward negative infinity due to the negative sign from \( -3 \).
- **Lack of a Finite Limit**: For a sequence to converge, it must settle into a specific, finite value. However, since \( a_n \) continues decreasing without bound, there is no such finite limit.
Thus, the sequence diverges because it fails to approach any specific value, illustrating the concept of divergence in sequences.
Other exercises in this chapter
Problem 30
Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be mor
View solution Problem 30
In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=1}^{\
View solution Problem 31
Use series to evaluate the limits. \begin{equation} \lim _{t \rightarrow 0} \frac{1-\cos t-\left(t^{2} / 2\right)}{t^{4}} \end{equation}
View solution Problem 31
Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{1+\ln n}\end{eq
View solution