Problem 27 - Precalculus: Mathematical for Calculus

Question

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{3}{n^{2}}\left[\frac{n(n+1)}{2}\right]$$

Step-by-Step Solution

Answer

The sequence is convergent with a limit of \( \frac{3}{2} \).

Step 1: Simplify the Expression Inside the Brackets

The given sequence is \( a_n = \frac{3}{n^2} \left[ \frac{n(n+1)}{2} \right] \). First, calculate the expression inside the brackets: \( \frac{n(n+1)}{2} \). This represents the sum of the first \( n \) positive integers, which simplifies to \( \frac{n^2 + n}{2} \).

Step 2: Simplify the Entire Expression

Now, substitute the simplified expression from Step 1 into the sequence definition: \[ a_n = \frac{3}{n^2} \left( \frac{n^2 + n}{2} \right) \]. Simplify this expression by distributing the division: \( a_n = \frac{3(n^2 + n)}{2n^2} \). Further simplify to get: \( a_n = \frac{3n^2 + 3n}{2n^2} = \frac{3}{2} + \frac{3}{2n} \).

Step 3: Analyze the Sequence for Convergence

As \( n \to \infty \), the term \( \frac{3}{2n} \to 0 \). Thus, \( a_n = \frac{3}{2} + \frac{3}{2n} \to \frac{3}{2} \). This means that the sequence converges.

Key Concepts

Limit of a Sequence

The limit of a sequence is a fundamental concept in mathematics, especially in calculus and analysis. It describes the behavior of a sequence as the index approaches infinity. Understanding limits helps in determining if a sequence converges to a specific value or not.
To find the limit of a sequence, we analyze the general term of the sequence. For the sequence given, \[ a_n = \frac{3}{n^2} \left( \frac{n^2 + n}{2} \right) \]we simplify it to \[ a_n = \frac{3}{2} + \frac{3}{2n} \]As the value of \( n \) becomes very large, the term \( \frac{3}{2n} \) approaches 0. Thus, the limit of the sequence \( a_n \) as \( n \) approaches infinity is \( \frac{3}{2} \).

This limit means that, over time, the terms of the sequence get arbitrarily close to \( \frac{3}{2} \). If the sequence reaches a unique finite limit, it is said that the sequence converges to this limit.

Convergent Sequence

A sequence is called convergent if its terms approach a specific finite number as the index tends to infinity. A convergent sequence has a limit, which we denote as \( L \), a real number. This concept is crucial for understanding how sequences behave in the long run.
For the particular sequence given by the formula:\[ a_n = \frac{3}{2} + \frac{3}{2n} \]we can observe its behavior for very large \( n \). As \( n \) increases without bound, the fraction \( \frac{3}{2n} \) decreases towards 0.
Therefore, the terms of the sequence \( a_n \) approach \( \frac{3}{2} \). This demonstrates that the sequence is convergent, with the limit being \( \frac{3}{2} \).

This convergence means for any small positive number, we can find a stage in the sequence where all subsequent terms are arbitrarily close to \( \frac{3}{2} \).

Simplifying Rational Expressions

Simplifying rational expressions is an important mathematical process often used to analyze the behavior of sequences and functions. It involves reducing expressions to their simplest form by canceling common factors and simplifying terms.
In the example problem, the objective was to simplify \[ a_n = \frac{3}{n^2} \left( \frac{n^2 + n}{2} \right) \]to a more manageable form. Recognizing that the expression inside the brackets \( \frac{n(n+1)}{2} \) can be expressed as \( \frac{n^2 + n}{2} \) allows easier manipulation.
Then, distributing the division yields \[ a_n = \frac{3(n^2 + n)}{2n^2} = \frac{3}{2} + \frac{3}{2n} \]Simplification involved breaking down the expression to terms that individually converge or diverge, helping in determining the sequence's limit.

Skills in simplifying rational expressions support better understanding of complex problems in calculus, algebra, and beyond. It's a strategy that paves the way for identifying limits and establishing convergence. The goal is to reach a form that reveals crucial properties of the function or sequence being studied.