Problem 17 - Thomas Calculus

Question

Which of the series in Exercises 13 46 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 \( p = 0.5 < 1 \).

Step 1: Identify the series type

The given series is \( \sum_{n=1}^{\infty} \frac{3}{\sqrt{n}} \). This is a p-series, which can be written as \( \sum_{n=1}^{\infty} n^{-p} \), with \( p = 0.5 \) in this case, since \( \frac{3}{\sqrt{n}} \) can be rewritten as \( 3n^{-0.5} \).

Step 2: Recall the p-series convergence test

A p-series \( \sum_{n=1}^{\infty} n^{-p} \) converges if \( p > 1 \) and diverges if \( p \leq 1 \). In the given series, the exponent \( p \) is 0.5, which is less than 1.

Step 3: Apply the convergence test

Since \( p = 0.5 \leq 1 \), according to the p-series test, the series \( \sum_{n=1}^{\infty} \frac{3}{\sqrt{n}} \) diverges.

Step 4: Conclude with the test result

The series does not meet the criteria for convergence as the p-value is less than 1. Therefore, the series diverges.

Key Concepts

p-series Test

The p-series test is a vital tool used in calculus to determine whether an infinite series converges or diverges. An infinite series is referred to as a p-series when it takes the form \[ \sum_{n=1}^{\infty} \frac{1}{n^p} \] where

• \( p \) is a positive constant.

To determine convergence using the p-series test, you can follow these simple rules: - If \( p > 1 \), then the p-series converges. - If \( p \leq 1 \), then the series diverges. In the given exercise, the series \( \sum_{n=1}^{\infty} \frac{3}{\sqrt{n}} \) is recognized as a p-series by re-writing it as \( 3n^{-0.5} \). Here, the exponent \( p = 0.5 \), and since \( 0.5 \leq 1 \), the series diverges by the p-series test. Understanding the exponent \( p \) is crucial as it directly influences the behavior of the infinite series.

Convergence

Convergence of an infinite series occurs when the sum of all its terms approaches a finite number as the number of terms becomes very large. In mathematical terms, we say that the series sum, when considered up to infinity, results in a finite limit. For any series \( \sum a_n \), if the series' partial sums \( S_N = a_1 + a_2 + \cdots + a_N \) tend to a definite number as \( N \to \infty \), the series is convergent.Some key facts about convergence include: - Convergent series do not "blow up" or approach infinity.- They often stabilize at some number, making them particularly useful in mathematical and real-world applications.In the context of p-series, if \( p > 1 \), the terms decrease fast enough that the series converges. For example, \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges because the exponent \( p = 2 \) is greater than 1.

Divergence

Divergence, in contrast to convergence, occurs when the sum of a series grows indefinitely as more terms are added. If a series does not meet the criteria for convergence, it is usually divergent. This can happen, for instance, if the series doesn't approach a finite limit, indicating the partial sums increase without bound.Generally, for the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \): - If \( p \leq 1 \), the series diverges. - When \( p = 1 \), it matches the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is a well-known example of a divergent series. In the exercise, \( \sum_{n=1}^{\infty} \frac{3}{\sqrt{n}} \) diverges because the term \( n^{-0.5} \) doesn't decrease fast enough for the series to approach a finite limit. Understanding divergence helps us recognize series that do not settle into a steady value, making them behave unpredictably over large sums.

Infinite Series

An infinite series is an important mathematical structure formed by adding up infinitely many terms. It takes the form \[ \sum_{n=1}^{\infty} a_n \] where

• \( a_n \) represents the general term of the series.

These series can either converge or diverge, and understanding which behavior they exhibit is key in various fields, including physics, engineering, and finance.Infinite series can be bewildering because they build upon the idea of summing an endless sequence of numbers. However, tools like the p-series test simplify them into manageable rules. For instance, recognizing that different values for \( p \) have such significant implications helps categorize infinite series efficiently.This concept is crucial in solving real-life problems and formulating models that aim to capture the essence of behavior that continues indefinitely. For example, calculating compound interest over time or understanding wave patterns are approximate representations tackled through infinite series. As such, they form an essential part of higher mathematics and theoretical studies.