Problem 59 - Essential Calculus

Question

In Exercises \(59-62,\) verify that the Ratio Test is inconclusive for the \(p\) -series. $$ \sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}} $$

Step-by-Step Solution

Answer

By applying the Ratio Test to the provided p-series, the limit of the absolute ratio of the (n+1)th term to the nth term, as n approaches infinity, is found to be 1. Therefore, the Ratio Test is inconclusive for this series.

Step 1: Identify the p-series

The given series is \(\sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}}\). In this series, 'p' is \(3 / 2\).

Step 2: Apply the Ratio Test

The Ratio Test requires to calculate the limit as n approaches infinity of the absolute ratio of the (n+1)th term and nth term of the series. So, calculate the limit as n goes to infinity of \(\left|\frac{\frac{1}{(n+1)^{3/2}})}{\frac{1}{n^{3/2}}}\right|\). This simplifies to \(\lim_{n \to \infty} \left| \frac{n^{3/2}}{(n+1)^{3/2}} \right|\).

Step 3: Evaluate the limit

To evaluate the limit \(\lim_{n \to \infty} \left| \frac{n^{3/2}}{(n+1)^{3/2}} \right|\), we can divide both the numerator and denominator by \(n^{3/2}\), which simplifies it to \(\lim_{n \to \infty} \left| \frac{1}{(1+1/n)^{3/2}} \right|\). As n approaches infinity, \(1/n\) approaches 0, so the limit can be simplified further to \(1\).

Step 4: Compare the limit with 1

The limit determined in the previous step is exactly 1. Therefore, according to the Ratio Test, the test is inconclusive about whether the series converges or diverges.

Key Concepts

Ratio Test

The Ratio Test is a popular method used to analyze the convergence of infinite series. It helps determine if a series converges, diverges, or if the test is inconclusive. To apply the Ratio Test, you calculate the limit of the ratio of the absolute values of successive terms. For a series \[ \sum_{n=1}^{\infty} a_n \] the Ratio Test involves evaluating: \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]

• If the limit is less than 1, the series converges.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

In the exercise, the limit was calculated to be 1, making the Ratio Test inconclusive for the given p-series. When the test is inconclusive, other methods are needed to determine the series' convergence.

p-series

A p-series is a type of series that takes the form: \[ \sum_{n=1}^{\infty} \frac{1}{n^p} \] where \( p \) is a real number. The behavior of a p-series depends on the value of \( p \). Here are the critical points to remember:

• If \( p > 1 \), the series converges.
• If \( p \leq 1 \), the series diverges.

The given series in our exercise, \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \), is a p-series with \( p = \frac{3}{2} \). Since \( \frac{3}{2} > 1 \), this series converges. However, the Ratio Test couldn't confirm this, which is why the test was inconclusive.

Convergence

Convergence is a property of a series where the sum of its terms approaches a finite value as more terms are added. For an infinite series to converge, its partial sums must tend toward a finite number. To check for convergence, various tests can be applied, such as the Ratio Test, the p-series test, the integral test, and others. Each has its conditions and applicability. In the case of the p-series \[ \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \] we know it converges because \( p = \frac{3}{2} > 1 \). Understanding convergence ensures that a sum won't approach infinity, providing valuable insight into the series' behavior and limits.

Infinite Series

An infinite series is the sum of infinitely many terms, written as \[ \sum_{n=1}^{\infty} a_n \] where \( a_n \) represents the terms of the series. Infinite series are essential in calculus, providing ways to represent functions, solve equations, and approximate values.Infinite series can converge or diverge:

• Convergent infinite series approach a finite sum.
• Divergent infinite series do not settle to a single value and often approach infinity.

Studying infinite series requires understanding their behavior over an infinite number of terms. Various tests, such as the Ratio Test, p-series test, and others, help determine the nature of the series. In the exercise's context, \[ \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \] is an example of a convergent infinite series, showing that even infinite processes can have finite results.