Problem 4 - Calculus: Early Transcendentals

Question

Calculate the sum of the series \( \sum_{n = 1}^{\infty} a_n \) whose partial sums are given. \( s_n = \frac {n^2 - 1}{4n^2 + 1} \)

Step-by-Step Solution

Answer

The sum of the series is \(\frac{1}{4}\).

Step 1: Understanding the Problem

We are given a series with partial sums defined as \(s_n = \frac{n^2 - 1}{4n^2 + 1}\). We need to calculate the sum \(\sum_{n=1}^{\infty} a_n\), where \(a_n = s_n - s_{n-1}\). To find the series sum, first calculate \(\lim_{n \to \infty} s_n\).

Step 2: Calculate Limit of Partial Sums

Calculate \(\lim_{n \to \infty} \frac{n^2 - 1}{4n^2 + 1}\). As \(n \to \infty\), this approaches \(\lim_{n \to \infty} \frac{1 - \frac{1}{n^2}}{4 + \frac{1}{n^2}} = \frac{1}{4}\).

Step 3: Finding the Sum of the Series

The sum of the series is given by \(\lim_{n \to \infty} s_n = \frac{1}{4}\). Therefore, the sum \(\sum_{n=1}^{\infty} a_n\) converges to \(\frac{1}{4}\).

Key Concepts

Partial Sums

When dealing with an infinite series, partial sums are crucial to understanding how the series behaves. The partial sum, denoted as \( s_n \), represents the sum of the first \( n \) terms of the series. In other words, if you have a series \( a_1 + a_2 + a_3 + \ldots \), the partial sum \( s_n \) is calculated as \( a_1 + a_2 + \ldots + a_n \). The goal is often to see what happens to these sums as \( n \) becomes very large.

• The partial sums give you a snapshot of the sum up to a certain number of terms.
• By examining \( s_n \), we can get insight into whether the series as a whole converges or diverges.
• In this case, each partial sum is defined as \( s_n = \frac{n^2 - 1}{4n^2 + 1} \).

This expression provides each new term by evaluating specific values of \( n \). Tracking these aids in the analysis needed to determine if they are approaching a particular value, indicating convergence.

Limit of a Function

The concept of a limit is fundamental in calculus and analysis. It helps us to understand the behavior of functions as they approach a certain point or value. In the context of infinite series, we often deal with limits of partial sums.To solve the original problem, we calculated \( \lim_{n \to \infty} s_n = \lim_{n \to \infty} \frac{n^2 - 1}{4n^2 + 1} \). Doing this involves recognizing the dominant terms in the numerator and denominator:

• As \( n \) becomes very large, terms like \(-1\) and \(+1\) become negligible compared to \(n^2\).
• Thus, the expression simplifies to \( \lim_{n \to \infty} \frac{1 - \frac{1}{n^2}}{4 + \frac{1}{n^2}} \).
• This simplifies further to \( \frac{1}{4} \) as \( \frac{1}{n^2} \to 0 \) when \( n \to \infty \).

Understanding limits in this way lets us predict the behavior of sequences and series as they progress towards infinity.

Convergence of Series

Convergence is a vital concept when working with infinite series. It tells us whether a series approaches a finite value or not. A series \( \sum_{n=1}^{\infty} a_n \) is said to converge to a value \( S \) if the sequence of its partial sums \( s_n \) approaches \( S \) as \( n \) tends to infinity.Here's where understanding partial sums and their limits becomes essential. Given the series' partial sums \( s_n = \frac{n^2 - 1}{4n^2 + 1} \), finding its limit as \( n \to \infty \) shows:

• If \( \lim_{n \to \infty} s_n = S \), then the series converges to \( S \).
• In our case, since \( \lim_{n \to \infty} s_n = \frac{1}{4} \), our series converges to \( \frac{1}{4} \).
• Convergence assures us of achieving a specific total, meaning the series has a well-defined sum.

This concept is crucial because only convergent series yield a finite, meaningful sum. Understanding the convergence of series is a key step in many mathematical applications!