Problem 50 - Precalculus with Limits

Question

In Exercises 49-52, find the limit of the sequence. Then verify the limit numerically by using a graphing utility to complete the table. $$ a_n = \dfrac{4}{n} \left(n + \dfrac{4}{n} \left[\dfrac{n(n+1)}{2} \right] \right) $$

Step-by-Step Solution

Answer

The limit of the sequence \( a_n \) as \( n \) approaches infinity is \( \infty \).

Step 1: Simplify the Sequence

First, the sequence needs to be simplified to a more manageable form. Start by simplification of the fraction inside the square brackets, this gives: \( a_n = \dfrac{4}{n} (n + 2n(n+1) ) \). Now, further simplifying the sequence results in \( a_n = 4 + 4(n+1) \).

Step 2: Find the Limit

Find the limit of \( a_n \) as \( n \) approaches infinity. In the case of \( a_n = 4 + 4(n+1) \), as \( n \) approaches infinity, only the \( 4(n+1) \) term is affected. The limit is \( \lim_{n\to\infty} (a_n) = \lim_{n\to\infty}(4 + 4(n+1)) = \infty \). This means that as \( n \) gets larger and larger, \( a_n \) gets larger without bound.

Step 3: Numerical Verification

Use a graphing tool to visualize the sequence and verify the limit. As you graph the sequence, you will see that the graph of \( a_n \) gets higher and higher as \( n \) increases, indicating again that the limit of this sequence as \( n \) approaches infinity is indeed \( \infty \).

Key Concepts

Limits of Sequences

The concept of limits is crucial when analyzing sequences. A sequence has a limit if its terms approach a specific value as the index becomes very large. This limit is what the sequence "settles down" to. However, not all sequences have a finite limit.
In our case, we have the sequence: \[a_n = 4 + 4(n+1)\]
To find its limit as \(n\) approaches infinity, consider how the terms behave: since \[4(n+1)\] tends to grow indefinitely large as \(n\) increases, \(a_n\) does not approach a finite value. Therefore, the limit of the sequence \(a_n\) as \(n\) approaches infinity is \(\infty\).
- A key takeaway is: **if the terms of a sequence grow without bound, then the limit is \(\infty\).**
This implies that such a sequence will continue to increase indefinitely as \(n\) becomes very large.

Sequence Simplification

Simplifying sequences helps us see patterns and establish limits more clearly. In the given exercise, simplifying the sequence involves breaking down the mathematical expression to understand its behavior.
Initially, the sequence was given as:\[a_n = \frac{4}{n} \left(n + \frac{4}{n} \left[\frac{n(n+1)}{2}\right]\right)\]
To simplify, the expression within the brackets is first simplified:- Start by calculating the inner product: \(\frac{n(n+1)}{2}\), making it easier to multiply by \(\frac{4}{n}\).- Through arithmetic simplification, it results in:- \[a_n = 4 + 4(n+1)\]
This form is much simpler and allows easier determination of the sequence's behavior as \(n\) grows.

• Simple expressions are easier to understand and to graph.
• Simplified sequences reveal patterns, aiding in predictions about limits.
• Breaking down complex expressions step by step can prevent errors.

Graphical Verification of Limits

Graphing is a practical approach to verifying the limits of sequences. It provides a visual representation of how a sequence behaves as \(n\) increases.
When graphing the sequence \(a_n = 4 + 4(n+1)\), you will observe that the points on the graph keep moving upwards, highlighting how \(a_n\) increases without bound as \(n\) becomes very large.
This graphical insight confirms:

• The sequence's limit is infinite because the values do not plateau at a certain height, but continue to rise.
• A graph can show trends clearly and help confirm predictions made from algebraic manipulation.
• Visual tools are especially useful when numerical or algebraic methods suggest that a sequence doesn't converge to a finite number.

Thus, graphs are not just tools for confirmation but can also aid in understanding the overall behavior of sequences, reinforcing our findings about limits.