Question

Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. $$\left\\{a_{n}\right\\}=\left\\{\frac{3 n}{\sqrt{n^{2}+1}}\right\\}$$

Short answer

The sequence converges to 3.

Step-by-step solution

Analyze the Sequence

Consider the sequence given by \( a_n = \frac{3n}{\sqrt{n^2 + 1}} \). To determine whether it converges or diverges, analyze the behavior of this sequence as \( n \) approaches infinity.

Simplify Expression for Large \(n\)

For large values of \( n \), the expression under the square root \( \sqrt{n^2 + 1} \) can be approximated by \( \sqrt{n^2} = n \) since the \( +1 \) becomes negligible. Thus, the sequence can be approximated by \( a_n = \frac{3n}{n} = 3 \).

Formulate the Limit

Based on the approximation for large \( n \), the sequence \( a_n = \frac{3n}{\sqrt{n^2 + 1}} \) can be rewritten as \( \lim_{n \to \infty} \frac{3n}{\sqrt{n^2 + 1}} = 3 \). Thus, the sequence approaches 3 as \( n \) becomes very large.

Conclusion

Since we've shown that the limiting expression approaches 3 as \( n \) goes to infinity, the sequence converges, and its limit is 3.

Key concepts

Limit of a Sequence

In mathematics, understanding the limit of a sequence is crucial to determining whether a sequence converges or diverges. A sequence is simply an ordered list of numbers, and the limit is what these numbers trend towards as we move further along the list. For instance, if we have the sequence \( a_n = \frac{3n}{\sqrt{n^2 + 1}} \), we're interested in what value \( a_n \) approaches as \( n \) becomes very large.

In our specific sequence, as \( n \) increases, the term \( \sqrt{n^2 + 1} \) can be closely approximated by \( \sqrt{n^2} = n \). This simplification helps us see that \( a_n \approx \frac{3n}{n} = 3 \). Thus, the sequence approaches the value 3. Therefore, we can say the limit of this sequence is 3 when \( n \to \infty \).

Understanding limits is essential because it gives us a tool to find the "end behavior" of the sequence, indicating whether the sequence has a particular destination or not.

Infinite Sequences

Infinite sequences are sequences that continue indefinitely, without an endpoint. They can either converge, meaning they settle towards a particular number, or diverge, meaning they stray further away from a single value as they progress.

To analyze infinite sequences, it's important to look at their general behavior. For the given sequence \( a_n = \frac{3n}{\sqrt{n^2 + 1}} \), we evaluated its behavior as \( n \) grows very large. This process, known as taking the limit, helps us determine the future behavior of an infinite sequence.

When dealing with infinite sequences, some helpful strategies include:

• Simplifying complex expressions.
• Comparing terms to identify dominant behaviors (in our case, \( n \) dominated \( n^2 + 1 \)).
• Observing patterns to determine convergence or divergence.

Understanding infinite sequences allows us to make meaningful statements about how sequences behave "in their entirety," which is particularly useful in calculus and advanced mathematics.

Sequence Divergence

Sequence divergence occurs when a sequence does not settle at a particular value as it progresses towards infinity. Instead of approaching a single number, the values of the sequence may oscillate or grow indefinitely. Divergence is the opposite of convergence and marks sequences that become erratic or continuously increase or decrease without stopping.

In our earlier example, if instead of simplifying to a constant, \( a_n = \frac{3n}{\sqrt{n^2 + 1}} \) had grown without bound or behaved unpredictably, it would have been classified as divergent. However, since it settled at 3, it did not diverge but rather converged.

To determine divergence, you can look for indicators such as:

• Infinite growth or decay: If terms keep getting larger or smaller without limit.
• Oscillation: If terms jump back and forth without trending towards a specific value.
• Lack of a limiting value: When no specific number adequately represents the sequence's end behavior.

Recognizing sequence divergence is a fundamental part of analyzing sequences, helping us to understand not just where a sequence settles, but whether it behaves predictably at all.