Problem 14 - Precalculus with Limits

Question

In Exercises 13-20, (a) rewrite the sum as a rational function \(S(n)\), (b) use \(S(n)\) to complete the table, and (c) find \(\lim_{n \to \infty} S(n)\). $$\displaystyle\sum_{i=1}^{n} \dfrac{i}{n^2}$$

Step-by-Step Solution

Answer

\[\lim_{n \to \infty} S(n) = \dfrac{1}{2}\]; rational function \(S(n) = \dfrac{n+1}{2n}\]

Step 1: Rewrite the sum as a rational function

The given sum can be rewritten as follows: \[S(n) = \dfrac{1}{n^2}\displaystyle\sum_{i=1}^{n} i = \dfrac{1}{n^2} \cdot \dfrac{n(n+1)}{2} = \dfrac{n(n+1)}{2n^2}\]

Step 2: Simplify the rational function

Simplify the above expression by cancelling \(n\) from the numerator and the denominator: \[S(n) = \dfrac{n+1}{2n}\]

Step 3: Use the rational function to complete the table

Here, values may vary depending on what values are needed for tables. Here are some examples: For \(n=1\), \(S(1) = 1\). For \(n=2\), \(S(2) = 1.5\). For \(n=3\), \(S(3) = 1.333\). For \(n=4\), \(S(4) = 1.25\).

Step 4: Find the limit of the function as n approaches infinity

We can now find the limit of \(S(n)\) as \(n\) approaches infinity. As \(n\) approaches infinity, the term +1 in the numerator becomes insignificant compared to \(n\), thus the limit of \(S(n)\) as \(n\) approaches infinity is \[\lim_{n \to \infty} S(n) = \lim_{n \to \infty} \dfrac{n+1}{2n} = \dfrac{1}{2}\]

Key Concepts

Understanding Limits in Rational Functions

A core concept in calculus is understanding how functions behave as variables approach certain points. In our case, we look at limits as the variable \( n \) approaches infinity.

For the rational function \( S(n) = \frac{n+1}{2n} \), the limit tells us the value that the function gets closer to as \( n \) becomes very large. When we talk about \( n \to \infty \), we essentially disregard terms that become negligible, like the \(+1\) in the numerator compared to \(n\).

After simplifying, we're just left with \( \frac{1}{2} \) as our steady solution when \(n\) grows indefinitely. Limits help us predict this behavior, offering insights into the long-term trends of functions.

Exploring Infinite Series

Infinite series are sequences of numbers where we sum terms to proceed towards a limit. The expression given, \( \sum_{i=1}^{n} \frac{i}{n^2} \), is a finite series that acts as a building block for understanding infinite series.

The finite sum provides an approximation as \( n \) scales up. To transform it into a rational function \( S(n) \), we summarize the relationships between terms, meaning we reduce the series into a more manageable form.

As \( n \) heads to infinity, the finite series begins to approximate an infinite one, allowing us to discuss limits effectively. Understanding how finite series approximate infinite ones is a key skill in mastering calculus.

Simplifying Expressions for Clarity

Simplifying complex expressions is a fundamental math skill, reducing complexity by eliminating unnecessary components. Initially, our sum \( \sum_{i=1}^{n} \frac{i}{n^2} \) seems intricate.

By rewriting it as \( S(n) = \frac{1}{n^2} \cdot \frac{n(n+1)}{2} \), we gain a clearer formula to evaluate. This simplification reduces our task to basic algebra: canceling terms and reorganizing components.

The resultant \( S(n) = \frac{n+1}{2n} \) is far more straightforward, revealing focus on significant parts of the function. Simplified expressions are easier to analyze and help when calculating limits or understanding series.