Problem 32 - Precalculus Mathematics for Calculus

Question

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{5}{n}\left(n+\frac{4}{n}\left[\frac{n(n+1)}{2}\right]\right)$$

Step-by-Step Solution

Answer

The sequence diverges.

Step 1: Simplify the Sequence Term

The term of the sequence is given by \( a_n = \frac{5}{n} \left(n + \frac{4}{n}\left[\frac{n(n+1)}{2}\right]\right) \). Start by simplifying inside the brackets:1. Expand the expression \( \frac{n(n+1)}{2} \), which simplifies to \( \frac{n^2 + n}{2} \).2. Substitute it back into the sequence: \( a_n = \frac{5}{n} \left(n + \frac{4}{n} \cdot \frac{n^2 + n}{2}\right) \).

Step 2: Attempt direct substitution

Try substituting the value the variable approaches directly into the expression.

Step 3: Handle indeterminate forms

If direct substitution gives an indeterminate form, apply L'Hopital's Rule, algebraic manipulation, or other techniques.

Step 4: Evaluate the limit

Compute the final value of the limit.

Step 5: State the conclusion

Express the final answer.

Key Concepts

Limit of a Sequence

To understand the limit of a sequence, we need to grasp what happens to the terms of the sequence as the index, usually denoted as \( n \), approaches infinity. The limit is essentially the value that the terms of the sequence approach as \( n \) becomes very large. If the sequence settles down to a particular number, we say it converges to that number, which is called the limit.

In the case of the sequence \( a_n = \frac{5}{n} \left(n + \frac{4}{n}\left[\frac{n(n+1)}{2}\right]\right) \), simplifying expressions can help us find this limit. After simplification, investigate the behavior as \( n \to \infty \) to determine convergence or divergence.

To find the limit:

• Simplify the sequence expressions.
• Factor out dominant terms.
• Evaluate as \( n \to \infty \).

For this sequence, each term simplifies, and by understanding how they behave as \( n \) grows, one can determine if the limit converges to a specific number or not.

Divergent Series

A divergent series occurs when the terms of a sequence do not settle down to any specific value as \( n \) increases. In other words, the terms keep changing wildly or keep getting larger without approaching a particular number.

To determine if a sequence diverges, look for patterns where the terms don't stabilize. For instance, if the sequence's terms get increasingly larger without bound, we label it as divergent.

In relation to the given sequence, determining divergence involves checking if the numerator or denominator or any expression within the term causes the term to grow without settling. Examine if any portion of the expression keeps increasing indefinitely as \( n \to \infty \).

Understanding divergence is crucial because it tells us that attempting to find a particular 'limit' would be futile as no such limit exists.

Simplifying Expressions

Simplifying expressions in sequences is crucial for understanding convergence or divergence. It often involves breaking down complex terms into more manageable parts to evaluate their behavior as \( n \) becomes large.

Steps to simplify:

• Simplify the inner parts first, like factors inside brackets or fractions inside fractions.
• Eliminate common terms or factors where possible.
• Look for dominant terms in the expressions that determine the sequence's behavior as \( n \to \infty \).

In our specific sequence \( a_n = \frac{5}{n}(n + \frac{4}{n}[\frac{n(n+1)}{2}]) \), simplifying helps unpack the behavior of the sequence's terms as \( n \) increases. Focus on cleaning up the fraction terms and reducing them to see if a pattern emerges.

Always remember, the goal of simplifying is to make it clear whether the sequence has a limit or diverges, by focusing on the most significant elements of the expression.