Question

For each of the following sequences, if the divergence test applies, either state that \(\lim _{n \rightarrow \infty} a_{n}\) does not exist or find \(\lim _{n \rightarrow \infty} a_{n} .\) If the divergence test does not apply, state why. \(a_{n}=\frac{n}{5 n^{2}-3}\)

Short answer

\( \lim_{n \to \infty} a_n = 0 \); divergence test not applicable.

Step-by-step solution

Identify the sequence

The given sequence is expressed as \( a_n = \frac{n}{5n^2 - 3} \). To determine if the divergence test applies, we'll first calculate the limit as \( n \to \infty \).

Calculate the limit of the sequence

To find \( \lim_{n \to \infty} a_n \), consider the expression \( \frac{n}{5n^2 - 3} \). As \( n \to \infty \) the term \( -3 \) becomes negligible, and the highest power of \( n \) in the denominator dominates. Thus, simplify it to \( \frac{n}{5n^2} = \frac{1}{5n} \).

Evaluate the simplified limit

Now evaluate \( \lim_{n \to \infty} \frac{1}{5n} \). As \( n \to \infty \), \( \frac{1}{n} \to 0 \). Therefore, \( \lim_{n \to \infty} \frac{1}{5n} = 0 \).

Determine applicability of the divergence test

The divergence test states if \( \lim_{n \to \infty} a_n eq 0 \), the series diverges. Here, \( \lim_{n \to \infty} a_n = 0 \), so the divergence test does not apply, but it doesn't mean the series converges. Further tests are needed for convergence.

Key concepts

Sequence Convergence

In mathematics, when we discuss sequence convergence, we are looking at whether the terms of a sequence approach a specific value as the sequence progresses. A sequence is considered convergent if its terms tend to get closer and closer to a certain number, known as the limit of the sequence, as the number of terms increases indefinitely. For instance, in the given exercise, we examined the sequence \(a_{n} = \frac{n}{5n^2 - 3}\). By calculating the limit, we determined that as \(n\) approaches infinity, the sequence converges to 0. This means that as you take more and more terms, the value of \(a_n\) becomes closer to 0.
It's crucial to understand convergence as it helps in determining the behavior of sequences and series in calculus. Notably, a sequence that does not converge to any real number is said to diverge.
Key points to remember:

• A sequence is convergent if it approaches a specific limit.
• The limit is the value the sequence terms get closer to as \(n\) becomes infinitely large.
• If no limit exists, the sequence is divergent.

Limits in Calculus

Limits are a fundamental concept in calculus and provide a way to precisely define continuity, derivatives, and integrals. In the context of sequences, limits allow us to analyze the behavior of the sequence as \(n\) increases without bound.
To calculate a limit of a sequence like \(a_{n} = \frac{n}{5n^2 - 3}\), it is essential to simplify the expression by comparing the highest degree terms in the numerator and denominator. This lets us intuitively understand how the sequence behaves. For example, we found that as \(n\) approaches infinity, the term \(-3\) becomes negligible, leading us to simplify the sequence to \(\frac{1}{5n}\). The limit of this simplified expression as \(n\) approaches infinity is 0.

Highlights of using limits in calculus:

• Limits help evaluate the behavior of a sequence as terms grow large.
• Simplifying terms often involves considering the highest power in polynomials.
• The concept extends beyond sequences to functions, derivatives, and integrals.

Series Divergence

Series divergence is linked to the behavior of sequences, specifically infinite series. An infinite series is simply the sum of an infinite sequence of terms. In the divergence test, also known as the nth-term test for divergence, we look for a non-zero limit of a sequence to indicate that the corresponding series diverges. If \(\lim_{n \to \infty} a_{n} eq 0\), then the series \(\sum a_{n}\) must diverge. However, if the limit is zero, this test is inconclusive, and further methods are required to determine convergence or divergence of the series.
For example, in our exercise, we found \(\lim_{n \to \infty} a_{n} = 0\). This result means the divergence test doesn't conclude whether the series converges or diverges. We would need to apply other tests like the ratio test, root test, or integral test, to get a definitive answer about convergence.
Important points on series divergence:

• The divergence test is quick but not always sufficient.
• If \(\lim_{n \to \infty} a_{n} eq 0\), the series definitely diverges.
• If the limit is zero, further tests are needed.

Infinite Sequences

Infinite sequences are sequences that continue indefinitely, with terms indexed by natural numbers. These sequences can have varying behaviors; they might converge, as in the sequence in our example, or diverge and not settle to any particular value.
Understanding the behavior of infinite sequences is essential since it forms the basis for defining and working with infinite series, limits, and many other mathematical constructs. An infinite sequence like our example involves elements like the numerator and high degree terms in the denominator that determine the sequence's behavior as it progresses.
When examining infinite sequences, consider:

• They extend indefinitely, requiring a method such as limits to analyze their behavior.
• Convergence and divergence are key characteristics to determine.
• The arithmetic structure guides the simplifying process to analyze sequence behavior for large values of \(n\).

These concepts are not only foundational in calculus but also are utilized in complex analysis, real analysis, and other advanced fields in mathematics.