Problem 19
Question
If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{1+n}{n+n^{2}}$$
Step-by-Step Solution
Verified Answer
The sequence is convergent and its limit is 0.
1Step 1: Identify the general term
The sequence is given by the general term \( a_n = \frac{1+n}{n+n^2} \). This term represents the sequence that depends on \( n \), which is the index of the sequence.
2Step 2: Express the general term in a simplified form
Simplify the expression by dividing both the numerator and the denominator by \( n^2 \), the highest power of \( n \) in the denominator: \[ a_n = \frac{\frac{1}{n^2} + \frac{n}{n^2}}{\frac{n}{n^2} + \frac{n^2}{n^2}} = \frac{\frac{1}{n^2} + \frac{1}{n}}{\frac{1}{n} + 1} \].
3Step 3: Evaluate the limit as \( n \to \infty \)
As \( n \to \infty \), the terms \( \frac{1}{n^2} \) and \( \frac{1}{n} \) both approach 0. Thus, the expression simplifies to \( \frac{0 + 0}{0 + 1} = 0 \).
4Step 4: Conclude about convergence
Since the limit of \( a_n \) is 0 as \( n \to \infty \), the sequence is convergent and its limit is 0.
Key Concepts
Convergence of SequencesNumerical SequencesAsymptotic Behavior
Convergence of Sequences
In mathematics, convergence is when the terms of a sequence approach a specific value, known as the limit, as you progress through the sequence. Every sequence has the potential to either converge or diverge. When a sequence converges, it means the values of the sequence get closer and closer to a particular number without ever going beyond it.
In the original exercise, our sequence given by \( a_n = \frac{1+n}{n+n^2} \) is analyzed for convergence by observing what happens to the sequence as \( n \) (which represents the term number) becomes very large. This is often noted as \( n \to \infty \).
To check convergence, simplifying and finding the limit is crucial. By dividing each component in the sequence's formula by the highest power found in the denominator, the expression becomes easier to evaluate as \( n \to \infty \). Once simplified, if the sequence approaches a constant value (in this case 0), it is said to converge. When a sequence converges, the number it approaches is called the limit of the sequence.
In the original exercise, our sequence given by \( a_n = \frac{1+n}{n+n^2} \) is analyzed for convergence by observing what happens to the sequence as \( n \) (which represents the term number) becomes very large. This is often noted as \( n \to \infty \).
To check convergence, simplifying and finding the limit is crucial. By dividing each component in the sequence's formula by the highest power found in the denominator, the expression becomes easier to evaluate as \( n \to \infty \). Once simplified, if the sequence approaches a constant value (in this case 0), it is said to converge. When a sequence converges, the number it approaches is called the limit of the sequence.
Numerical Sequences
Numerical sequences are ordered lists of numbers, each identified by its position in the list. Each number in the sequence is called a term and can be represented by a general formula. In the exercise, the sequence \( a_n = \frac{1+n}{n+n^2} \) illustrates how numbers can be systematically generated based on their position \( n \).
Understanding how sequences are constructed and represented is fundamental in analyzing their behavior. By substituting different values of \( n \) into the sequence formula, you get different terms. For example, for \( n=1 \), the term is \( \frac{2}{2} = 1 \) and for \( n=2 \), the term is \( \frac{3}{6} = \frac{1}{2} \).
Sequences can portray growing, shrinking, or constant patterns depending on their mathematical formula. Recognizing these patterns helps determine how sequences behave over a long duration.
Understanding how sequences are constructed and represented is fundamental in analyzing their behavior. By substituting different values of \( n \) into the sequence formula, you get different terms. For example, for \( n=1 \), the term is \( \frac{2}{2} = 1 \) and for \( n=2 \), the term is \( \frac{3}{6} = \frac{1}{2} \).
Sequences can portray growing, shrinking, or constant patterns depending on their mathematical formula. Recognizing these patterns helps determine how sequences behave over a long duration.
Asymptotic Behavior
Asymptotic behavior looks at how sequences behave as the terms go to infinity (i.e., very large values). It provides insights into the boundary behavior of sequences, allowing us to describe what happens to the sequences without precisely calculating every term.
The asymptotic behavior gives us a means to describe sequences in a simplified way, especially when dealing with limits. As seen in the solution's simplification of \( a_n = \frac{\frac{1}{n^2} + \frac{1}{n}}{\frac{1}{n} + 1} \), the terms \( \frac{1}{n^2} \) and \( \frac{1}{n} \) become negligible as \( n \to \infty \), highlighting that the sequence approaches \( 0 \).
This insight shows that despite the complexity of the sequence's formula, the values become small and approach a limit. Observing asymptotic behavior not only aids in understanding where a sequence ultimately leads, but it also simplifies complex analyses by focusing on the most influential terms for large \( n \). This is essential in many fields of mathematics and helps understand real-world phenomena.
The asymptotic behavior gives us a means to describe sequences in a simplified way, especially when dealing with limits. As seen in the solution's simplification of \( a_n = \frac{\frac{1}{n^2} + \frac{1}{n}}{\frac{1}{n} + 1} \), the terms \( \frac{1}{n^2} \) and \( \frac{1}{n} \) become negligible as \( n \to \infty \), highlighting that the sequence approaches \( 0 \).
This insight shows that despite the complexity of the sequence's formula, the values become small and approach a limit. Observing asymptotic behavior not only aids in understanding where a sequence ultimately leads, but it also simplifies complex analyses by focusing on the most influential terms for large \( n \). This is essential in many fields of mathematics and helps understand real-world phenomena.
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