Problem 23 - Calculus: Early Transcendentals

Question

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {5 + 2n}{(1 + n^2)^2} \)

Step-by-Step Solution

Answer

The series converges.

Step 1: Understand the Series

The given series is \( \sum_{n = 1}^{\infty} \frac{5 + 2n}{(1 + n^2)^2} \). We need to determine if this series converges or diverges.

Step 2: Apply Limit Comparison Test

To apply the Limit Comparison Test, compare the terms of the series \( u_n = \frac{5 + 2n}{(1 + n^2)^2} \) with the terms \( a_n = \frac{n}{n^4} = \frac{1}{n^3} \) of a known convergent or divergent series. The series \( \sum \frac{1}{n^3} \) is known to converge because it is a p-series with \( p = 3 > 1 \).

Step 3: Compute the Limit

Compute \( \lim_{{n \to \infty}} \frac{u_n}{a_n} = \lim_{{n \to \infty}} \frac{\frac{5 + 2n}{(1 + n^2)^2}}{\frac{1}{n^3}} = \lim_{{n \to \infty}} \frac{(5 + 2n) n^3}{(1 + n^2)^2} \). Factor and simplify:\[ \lim_{{n \to \infty}} \frac{n^3 (5 + 2n)}{n^4} = \lim_{{n \to \infty}} \frac{5n^3 + 2n^4}{n^4} = \lim_{{n \to \infty}} \left(\frac{5}{n} + 2\right) = 2 \]

Step 4: Interpret the Limit

The limit \( \lim_{{n \to \infty}} \frac{u_n}{a_n} = 2 \) is a positive finite number. Thus, according to the Limit Comparison Test, both series \( \sum u_n \) and \( \sum a_n \) either converge or diverge simultaneously.

Step 5: Conclude the Test

Since the series \( \sum a_n = \sum \frac{1}{n^3} \) converges and the Limit Comparison Test implies \( \sum u_n \) behaves like \( \sum a_n \), the given series \( \sum \frac{5 + 2n}{(1 + n^2)^2} \) also converges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a valuable tool when analyzing the convergence or divergence of an infinite series. It involves comparing your series with another series whose convergence is already known.
Compare the terms of your series with the terms of a well-known series. Then, compute the limit of the quotient of the two series' terms as the variable approaches infinity.

• If the limit is a positive finite number, both series will either converge or diverge together.
• If the limit is zero or infinity, the test is inconclusive. You will need to use another method.

For our series, we employed this test by comparing our series to a convergent p-series, simplifying the quotient, and determining the limit to be 2. As this is a positive finite number, we can conclude the behavior of the original series.

Convergent Series

A convergent series is a series whose sum approaches a specific finite number as more and more terms are added.
When a series converges, it means that regardless of how many terms you add up, the total will never exceed (or go below) a certain point. This is a critical concept for determining the behavior of infinite series.

• The most common way to determine convergence is using tests like the Limit Comparison Test, Ratio Test, or Integral Test.
• Convergence can signify that a model or real-world phenomena is stable or predictable.

In our example, the series \( \sum \frac{1}{n^3} \) is tested and is known to converge, achieving a finite sum through known standard convergence tests for series.

Divergent Series

A divergent series is the opposite of a convergent series. It either increases without bound or fails to approach a particular value as more terms are added.
This infers that the sum of the series continues to grow infinitely without settling into a specific finite number.

• Divergence can occur when a series' terms do not decrease quickly enough as the index increases.
• Some series naturally diverge because their terms do not trend towards zero.

Identifying if a series is divergent is crucial for understanding the limits of what certain mathematical models can predict or represent. If in analysis, the series were indeed found to diverge, other modeling approaches would be reconsidered.

P-Series

P-Series is a fundamental type of series that is highly prevalent in convergence tests. Its general form is \( \sum \frac{1}{n^p} \), where \( p \) is a constant.
Understanding the convergence behavior of a p-series is essential because it directly affects the limit comparison test outcomes when assessing other series.

• If \( p > 1 \), the p-series converges, meaning its terms decrease rapidly enough.
• If \( p \leq 1 \), the series diverges.

In the original problem, the series \( \sum \frac{1}{n^3} \) is a convergent p-series with \( p = 3 \). This characteristic helped us determine the convergence of our main series using the Limit Comparison Test.