Problem 9
Question
Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\frac{n^{3}}{n^{4}+1}\right\\}$$
Step-by-Step Solution
Verified Answer
Answer: The limit of the sequence as \(n\) approaches infinity is 0.
1Step 1: Identify the important terms
Observe the sequence:
$$a_n = \frac{n^{3}}{n^{4}+1}$$
The important terms are numerator \(n^3\) and denominator \(n^4+1\).
2Step 2: Consider the highest power terms
In both numerator and denominator, consider the terms with the highest power of \(n\) which are \(n^3\) and \(n^4\).
3Step 3: Divide numerator and denominator by the highest power of n in the denominator
We will divide both the numerator and the denominator by \(n^4\):
$$a_n = \frac{\frac{n^3}{n^4}}{\frac{n^4 + 1}{n^4}}$$
4Step 4: Simplify expressions in the numerator and denominator
Simplify the fraction to get:
$$a_n = \frac{\frac{1}{n}}{1 + \frac{1}{n^4}}$$
5Step 5: Determine the limit as n approaches infinity
As \(n\) approaches infinity, the terms \(\frac{1}{n}\) and \(\frac{1}{n^4}\) both approach 0. This means that the limit of the sequence is:
$$\lim_{n\to\infty} a_n = \frac{0}{1+0} = 0$$
Thus, the limit of the sequence as \(n\) approaches infinity is 0.
Key Concepts
ConvergenceSequencesAsymptotic Behavior
Convergence
Understanding convergence is essential to determining the limit of a sequence. When we talk about a sequence "converging," we mean that as the sequence progresses (or as the term number "n" increases), its terms get closer and closer to a specific value. This specific value is what we refer to as the "limit." If a sequence converges, it implies the terms settle down to that limit.
A great way to conceptualize this is to think of a sequence like a car approaching a stop sign. As it gets closer, it slows until it comes to a complete stop at the sign, which symbolizes its limit. In the context of our original sequence, \(a_n = \frac{n^3}{n^4+1}\), it approaches the limit of zero as the n gets larger. In simpler terms:
A great way to conceptualize this is to think of a sequence like a car approaching a stop sign. As it gets closer, it slows until it comes to a complete stop at the sign, which symbolizes its limit. In the context of our original sequence, \(a_n = \frac{n^3}{n^4+1}\), it approaches the limit of zero as the n gets larger. In simpler terms:
- The difference between the terms of the sequence and their limit becomes negligible as n grows.
- Convergence tells us that the sequence won't keep changing its course unpredictably but instead will settle at zero.
Sequences
Sequences are ordered lists of numbers, often following a specific rule to generate each term. In mathematics, sequences like \( a_n = \frac{n^3}{n^4+1} \) involve letting "n" be any positive integer to find subsequent terms. Each number in a sequence is called an element or a term.
There are infinite and finite sequences:
When dealing with more complex sequences, it helps to focus on the dominant terms, which are the terms with the highest powers of "n." For our sequence, these are \ n^3 \ in the numerator and \ n^4 \ inside the denominator. Then, by simplifying, we can see how the sequence behaves.
There are infinite and finite sequences:
- Finite sequences have a specific number of terms. They stop at a certain point.
- Infinite sequences go on forever, much like the one we're examining here.
When dealing with more complex sequences, it helps to focus on the dominant terms, which are the terms with the highest powers of "n." For our sequence, these are \ n^3 \ in the numerator and \ n^4 \ inside the denominator. Then, by simplifying, we can see how the sequence behaves.
Asymptotic Behavior
Asymptotic behavior gives us insight into how a sequence behaves at the extremes, as n increases without bound. A sequence's asymptotic nature tells us about its long-term trends, allowing us to approximate the sequence as n tends to infinity. This is crucial because it helps us predict where the sequence is heading.
In our example sequence \( a_n = \frac{n^3}{n^4+1} \), we can see that as n becomes extremely large, the little numbers become less significant, and the sequence can be approximated more simply.
Asymptotic analysis often involves dividing the terms of the sequence by the largest power of "n" found in the denominator, which simplifies the expression and shows what's happening at infinity. Here:
In our example sequence \( a_n = \frac{n^3}{n^4+1} \), we can see that as n becomes extremely large, the little numbers become less significant, and the sequence can be approximated more simply.
Asymptotic analysis often involves dividing the terms of the sequence by the largest power of "n" found in the denominator, which simplifies the expression and shows what's happening at infinity. Here:
- We divided each term by \ n^4 \ to focus on how the leading terms affect the behavior.
- Knowing that \ \frac{1}{n} \ and \ \frac{1}{n^4} \ both go to zero helps us see that the sequence converges to zero.
Other exercises in this chapter
Problem 9
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. $$\sum_{k=0}^{\infty} \frac{k}{2 k+1}$$
View solution Problem 9
Evaluate each geometric sum. $$\sum_{k=0}^{20}\left(\frac{2}{5}\right)^{2 k}$$
View solution Problem 9
Write the first four terms of the sequence \(\left\\{a_{n}\right\\}_{n=1}^{\infty}\) $$a_{n}=1 / 10^{n}$$
View solution Problem 10
Give an example of a series that converges conditionally but not absolutely.
View solution