Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use any test developed so far, including any from Section \(9.2\), to decide about the convergence or divergence of the series. Give a reason for your conclusion. $$ \sum_{k=1}^{\infty} \frac{1}{1+4 k^{2}} $$

Step-by-Step Solution

Answer

The series converges by the Limit Comparison Test with a \( p \)-series.

Step 1: Identify the Series Type

The given series is \( \sum_{k=1}^{\infty} \frac{1}{1+4k^2} \). Notice the series resembles a form of a \( p \)-series with \( a_k = \frac{1}{1+4k^2} \), which is hyperbolic or related to an integral test due to the presence of a quadratic term in the denominator.

Step 2: Choose a Convergence Test

Since the series is similar to the form \( \frac{1}{1+k^2} \), which looks like \( \frac{1}{k^2} \), we can try a comparison test or use the Limit Comparison Test to establish convergence properties.

Step 3: Simplify the Expression

Re-examine the function \( b_k = \frac{1}{4k^2} \). Let's consider the series \( \sum_{k=1}^{\infty} \frac{1}{4k^2} \), which converges because it is a \( p \)-series with \( p=2 > 1 \).

Step 4: Apply the Limit Comparison Test

Apply the Limit Comparison Test with the convergent \( p \)-series \( b_k = \frac{1}{4k^2} \):\[\lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} \frac{1+4k^2}{4k^2} = \lim_{k \to \infty} \frac{1}{4k^2} + 1 = 1.\]Since the limit is positive and finite, the Limit Comparison Test confirms \( a_k \) converges because \( b_k \) converges.

Step 5: Conclude Convergence

Since the Limit Comparison Test shows that the original series behaves like a convergent \( p \)-series, the series \( \sum_{k=1}^{\infty} \frac{1}{1+4k^2} \) converges.

Key Concepts

p-series

A p-series is a type of infinite series where each term has the form \(\frac{1}{k^p}\), where \(k\) is a natural number and \(p\) is a constant greater than zero. This series is significant because its convergence largely depends on the value of \(p\).

• If \(p > 1\), the p-series converges, meaning that its sum approaches a finite number as the number of terms goes to infinity.
• If \(p \leq 1\), the series diverges, meaning its sum does not approach a finite number.

In the context of the given problem, we are dealing with a series that can be compared to a p-series with \(p = 2\), which is known to converge. Recognizing the p-series helps simplify the analysis of more complex series by allowing comparison to simple, well-known series.

limit comparison test

The Limit Comparison Test is a helpful tool in the study of infinite series when direct comparison is challenging. It is used to determine the convergence or divergence of a series by comparing it to another series with known behavior.Here's how it works:

• Suppose \(a_k\) and \(b_k\) are series with positive terms.
• Evaluate \(\lim_{k \to \infty} \frac{a_k}{b_k}\).
• If the limit is a positive finite number, then \(a_k\) converges if \(b_k\) converges, and diverges if \(b_k\) diverges.

For the given series \(\sum_{k=1}^{\infty} \frac{1}{1+4k^2}\), the Limiting Comparison Test was applied with \(b_k = \frac{1}{4k^2}\), a convergent p-series. The limit calculation showed that the given series behaves similarly to this convergent series, thus confirming its convergence.

infinite series

Infinite series involve the summation of an infinite sequence of terms, often represented as \(\sum_{k=1}^{\infty} a_k\). The central question for an infinite series is whether it converges or diverges.

• A series converges if the sequence of partial sums approaches a specific value, called the sum of the series.
• If the sequence of partial sums does not approach a finite number, the series diverges.

In mathematics, the convergence of infinite series is evaluated using various tests, including p-series test and comparison tests. The original series \(\sum_{k=1}^{\infty} \frac{1}{1+4k^2}\) is an example where convergence was determined using these techniques. Understanding infinite series and their behavior is fundamental in calculus and mathematical analysis, providing insights into complex mathematical and real-world phenomena.