Problem 15 - Thomas Calculus

Question

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{3}{\sqrt{n}} $$

Step-by-Step Solution

Answer

The series diverges because it is a p-series with \( p = \frac{1}{2} < 1 \).

Step 1: Identify the Type of Series

The given series is \( \sum_{n=1}^{\infty} \frac{3}{\sqrt{n}} \). This is a p-series in the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) where \( p = \frac{1}{2} \).

Step 2: Determine Convergence or Divergence of a p-Series

For a p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), it converges if \( p > 1 \) and diverges if \( p \leq 1 \). Here, \( p = \frac{1}{2} \), which is less than 1.

Step 3: Conclusion Based on p-Series Test

Since \( p = \frac{1}{2} \) is less than 1, the given series \( \sum_{n=1}^{\infty} \frac{3}{\sqrt{n}} \) diverges by the p-series test.

Key Concepts

Series Convergence

When studying infinite series, a key question is whether a series converges or diverges. Convergence, in simple terms, means that adding up an infinite number of terms gets you closer and closer to a specific number. Imagine a series as a long trail of breadcrumbs. If they converge, they are leading you towards a specific destination.
If they diverge, the trail keeps going, and you never land anywhere specific.
Understanding convergence is crucial because it helps us determine if the infinite series has a sum that can be quantified. This is common in calculus and mathematical analysis and is applicable in real-world scenarios ranging from engineering to economics.

• If a series converges, we can often find its sum or describe its behavior.
• If a series diverges, its sum does not settle to a particular value and continues infinitely.

When trying to determine the convergence of a series, it is helpful to recognize the type of series you are dealing with, like a p-series, geometric series, or others. This recognition can often lead directly to the answer, as it did in our problem with the p-series.

P-Series

P-series are a common type of series that are easy to recognize and analyze. They take the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive number. These are fascinating because whether they converge or diverge depends entirely on the value of \( p \).

• If \( p > 1 \), the series converges. Each term gets small enough fast enough for the entire sum to settle on a single value.
• If \( p \leq 1 \), the series diverges, as the terms do not decrease quickly enough to sum to a finite value.

In our exercise, the series given is \( \sum_{n=1}^{\infty} \frac{3}{\sqrt{n}} \). This is indeed a p-series, especially when simplified to \( \sum_{n=1}^{\infty} \frac{3}{n^{1/2}} \). Here, \( p = \frac{1}{2} \), clearly showing us that this series fits the criteria for divergence in a p-series since \( 1/2 \leq 1 \).
Recognizing this pattern, we immediately see that we do not need to calculate further; the series diverges.

Divergence Test

When faced with an infinite series, one of the quick ways to test if it diverges is the divergence test, also known as the nth-term test for divergence. This test is simple: if the limit of the a_n term as \( n \to \infty \) is not zero, the series by default diverges.
This test can quickly determine inconclusive cases where other tests might be too involved or complex, but it has its limitations.

• If the limit of a_n is zero, the series may converge or diverge, requiring further testing.
• If the limit is not zero, the series definitively diverges.

However, for many series, like the p-series example we discussed, divergence is straightforwardly established using the specifics of the p-series test, rendering the basic divergence test unnecessary. But, knowing and applying various tests gives us the power to solve a range of problems efficiently.