Problem 34 - Precalculus: Mathematics for Calculus

Question

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{12}{n^{4}}\left[\frac{n(n+1)}{2}\right]^{2}$$

Step-by-Step Solution

Answer

The sequence is convergent with a limit of 3.

Step 1: Identify the Sequence Terms

Given the sequence \(a_n = \frac{12}{n^4}\left(\frac{n(n+1)}{2}\right)^2\), let's first rewrite the expression by simplifying the brackets. The term inside the brackets is the square of the sum of the first \(n\) natural numbers, which is \(\frac{n(n+1)}{2}\).

Step 2: Simplify the Expression

Simplify the term: \[ \left( \frac{n(n+1)}{2} \right)^2 = \left( \frac{n^2 + n}{2} \right)^2 = \frac{(n^2 + n)^2}{4}. \] Substitute back into the sequence expression: \[ a_n = \frac{12}{n^4} \cdot \frac{(n^2 + n)^2}{4} = \frac{12(n^4 + 2n^3 + n^2)}{4n^4}. \]

Step 3: Further Simplify and Divide Terms

Now, simplify the expression: \[ a_n = \frac{12}{4} \cdot \left(1 + \frac{2}{n} + \frac{1}{n^2}\right) = 3 \left(1 + \frac{2}{n} + \frac{1}{n^2}\right). \]

Step 4: Evaluate the Limit as n Approaches Infinity

Consider the limit as \(n\) approaches infinity: \[ \lim_{{n \to \infty}} a_n = \lim_{{n \to \infty}} 3 \left(1 + \frac{2}{n} + \frac{1}{n^2}\right). \] As \(n\) becomes very large, \(\frac{2}{n}\) and \(\frac{1}{n^2}\) both approach zero. Therefore, \[ \lim_{{n \to \infty}} a_n = 3 \cdot (1 + 0 + 0) = 3. \]

Step 5: Conclusion on Convergence and Limit

Since the limit exists, the sequence \(a_n\) is convergent, and the limit is \(3\).

Key Concepts

Convergence and Divergence

The concepts of convergence and divergence are pivotal in understanding the behavior of sequences as they progress toward infinity. In simple terms:

• Convergence occurs when a sequence approaches a particular value as the number of terms goes to infinity. This means that, after a certain point, all terms of the sequence get arbitrarily close to the limit value.


• Divergence is characterized by a sequence that does not settle towards any specific number, even as it extends infinitely. Such a sequence continues to grow, shrink, or oscillate endlessly without nearing a single value.

To determine whether a sequence is convergent, one needs to find the limit as the number of terms, often denoted as \( n \), approaches infinity. If this limit exists and is a real number, the sequence converges to that number, ensuring predictability and stability in its behavior as \( n \) gets very large.

Sequence Expressions

A sequence is essentially a list of numbers following a specific rule or formula. A sequence expression is the equation that denotes how to generate the terms of this sequence. In the given exercise, the sequence term is expressed as:\[ a_{n} = \frac{12}{n^4} \left( \frac{n(n+1)}{2} \right)^2.\]Here, the sequence combines basic arithmetic operations and power rules.

• The numerator \( n(n+1)/2 \) represents the sum of the first \( n \) natural numbers divided by 2, a calculation often seen in arithmetic series.


• The term is then squared, indicating we're taking the square of this sum to incorporate quadratic growth into the sequence's pattern.


• The entire expression is divided by \( n^4 \), normalizing the term and modeling how the sequence decreases with the large power of \( n \) in the denominator.

Understanding sequence expressions involves breaking down these components to predict the behavior of the sequence as \( n \) increases.

Limit Evaluation

Limit evaluation is a core technique used to analyze the behavior of sequences and determine their convergence. By evaluating the limit of a sequence's term as \( n \) grows infinitely large, one can ascertain the sequence's long-term trend.In the exercise solution:

• First, simplify the given sequence expression to make limit calculations easier: \( a_{n} = 3(1 + \frac{2}{n} + \frac{1}{n^2}) \).


• Evaluate \( \lim_{n \to \infty} a_{n} \). Here, as \( n \) becomes very large, the fractions \( \frac{2}{n} \) and \( \frac{1}{n^2} \) shrink towards zero. This reflects the diminishing impact of these terms on the sequence.


• Thus, the expression simplifies to 3, since all remaining contributions except 1 vanish.

Through this evaluative process, we not only find that the sequence converges, but also determine the specific value it approaches: the limit, which is 3 in this instance.