Problem 12 - Calculus: Early Transcendentals

Question

Test the series for convergence or divergence. \( \displaystyle \sum_{k = 1}^{\infty} \frac {1}{k \sqrt{k^2 +1}} \)

Step-by-Step Solution

Answer

The series \( \sum_{k=1}^{\infty} \frac{1}{k \sqrt{k^2+1}} \) converges.

Step 1: Choose a Test for Convergence

To determine the convergence of the series, we can use the Limit Comparison Test, as it involves a series of terms with a rational expression, which is typical for such tests.

Step 2: Compare with a Known Series

Consider the series \( \sum_{k=1}^{\infty} \frac{1}{k^2} \), which is a known convergent p-series with \( p = 2 \). We will compare our series to this using the limit comparison test.

Step 3: Apply the Limit Comparison Test

Calculate the limit of the ratio of the given series term to the term of the comparison series:\[L = \lim_{k \to \infty} \frac{1 / (k \sqrt{k^2 + 1})}{1 / k^2} = \lim_{k \to \infty} \frac{k^2}{k \sqrt{k^2 + 1}}\]Simplifying this expression gives:\[L = \lim_{k \to \infty} \frac{k}{\sqrt{k^2 + 1}} = \lim_{k \to \infty} \frac{1}{\sqrt{1 + \frac{1}{k^2}}}\]As \(k\) approaches infinity, \(\frac{1}{k^2}\) approaches 0, so:\[L = \frac{1}{\sqrt{1 + 0}} = 1\]

Step 4: Conclude with the Limit Comparison Test

Since the limit \(L = 1\) is a finite positive number, and \(\sum_{k=1}^{\infty} \frac{1}{k^2}\) is a convergent series, by the Limit Comparison Test, the original series \( \sum_{k = 1}^{\infty} \frac {1}{k \sqrt{k^2 +1}} \) also converges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a method used to determine the convergence or divergence of an infinite series by comparing it to another series that is already known to converge or diverge. This test is particularly useful when dealing with series that have complicated terms. The idea is to:

• Select a known series which resembles the complicated series.
• Calculate the limit of the ratio of the terms from both series as the index approaches infinity.

If this limit is a positive finite number, the two series behave the same way regarding convergence or divergence. For example, consider the series \[ \sum_{k=1}^{\infty} \frac{1}{k^2} \]which is a known p-series that converges. When applying the Limit Comparison Test to \[ \sum_{k = 1}^{\infty} \frac {1}{k \sqrt{k^2 +1}} \]and comparing it to \( \sum_{k=1}^{\infty} \frac{1}{k^2} \), the limit of the ratio of their terms as \( k \to \infty \) is 1, thus confirming convergence.

P-Series

A p-series is a type of infinite series that has the form:\[ \sum_{k=1}^{\infty} \frac{1}{k^p} \]where \( p \) is a positive real number. The convergence of a p-series depends on the value of \( p \):

• The series converges if \( p > 1 \).
• The series diverges if \( p \leq 1 \).

This simple rule allows us to easily identify the convergence of many series. For example, the series \[ \sum_{k=1}^{\infty} \frac{1}{k^2} \]is a p-series with \( p = 2 \), which converges because \( 2 > 1 \). This knowledge is powerful when analyzing more complex series using tests like the Limit Comparison Test.

Series Convergence

Series convergence refers to whether the sum of an infinite series converges to a finite limit or not. Convergence tests are the various methods used to determine the behavior of these series. It's essential for mathematical analysis because:

• If a series converges, its terms approach 0 as the index tends to infinity.
• If a series diverges, the sum grows unbounded or does not settle to a single value.

Understanding convergence is crucial in fields such as calculus and real analysis, where series are used to represent functions, solve differential equations, and approximate values. Infinite series, like the geometric series or the p-series, have criteria that help us quickly determine their convergence.

Infinite Series

An infinite series is the sum of an infinite sequence of terms. Infinite series can be expressed in the form:\[ a_1 + a_2 + a_3 + \cdots = \sum_{k=1}^{\infty} a_k \]Infinite series are generally classified into different types, including geometric, arithmetic, harmonic, and p-series. Understanding infinite series is fundamental because it allows us to:

• Model natural phenomena and apply them in scientific computations.
• Explore different approximation techniques and numerical analysis methods.
• Make predictions and verify asymptotic properties of functions.

The convergence properties of infinite series shed light on their applications and potential limitations. Knowing when and how an infinite series converges is essential in tackling complex problems in various mathematical contexts.