Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{n+3}{n^{2} \sqrt{n}} $$

Step-by-Step Solution

Answer

The series converges by the p-series test.

Step 1: Simplify the Series Terms

Consider the series \( \sum_{n=1}^{\infty} \frac{n+3}{n^{2} \sqrt{n}} \). First, simplify the terms: \( \frac{n+3}{n^{2} \sqrt{n}} = \frac{n}{n^{2} \sqrt{n}} + \frac{3}{n^{2} \sqrt{n}} \). This simplifies to \( \frac{1}{n^{3/2}} + \frac{3}{n^{5/2}} \).

Step 2: Compare with a Known Series

Decompose the simplified terms into two separate series: \( \sum \frac{1}{n^{3/2}} \) and \( \sum \frac{3}{n^{5/2}} \). Recognize these as p-series, where \( p = 3/2 \) and \( p = 5/2 \), respectively.

Step 3: Determine Convergence of the Series

For a p-series \( \sum \frac{1}{n^p} \), the series converges if \( p > 1 \). Here, both \( \frac{1}{n^{3/2}} \) and \( \frac{3}{n^{5/2}} \) have \( p > 1 \). Therefore, both series converge.

Step 4: Conclusion of the Series' Convergence

Since both the series \( \sum \frac{1}{n^{3/2}} \) and \( \sum \frac{3}{n^{5/2}} \) converge, their sum, \( \sum \left( \frac{1}{n^{3/2}} + \frac{3}{n^{5/2}} \right) \), also converges.

Key Concepts

p-series test

The p-series test is a well-known method used to determine the convergence or divergence of a series. A p-series takes the form of \( \sum \frac{1}{n^p} \). Here, \( p \) is a positive constant that plays a crucial role in assessing whether the series converges or diverges.
For convergence of a p-series, the value of \( p \) must be greater than 1. If \( p > 1 \), the series converges. If \( p \leq 1 \), the series diverges.

• Example: \( \sum \frac{1}{n^2} \) converges because \( p = 2 \), which is greater than 1.
• In contrast, \( \sum \frac{1}{n} \) diverges as \( p = 1 \), which does not satisfy the convergence condition.

The p-series test is particularly useful because it can be applied directly to many common series, making it a handy tool in your mathematical toolkit.

mathematical series

A mathematical series is simply the sum of the terms of a sequence. In more formal terms, it is written as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the individual terms of the sequence.
Series come in many forms, such as geometric series, harmonic series, or the ones discussed here: p-series. The calculation of their convergence or divergence is a fundamental part of calculus.
Elements of understanding series include:

• Recognition of the type of series, whether it's geometric, harmonic, etc.
• Application of appropriate tests like p-series test, ratio test, or comparison test among others.
• Interpretation of series behavior as \( n \) approaches infinity, which directly affects convergence.

Mathematical series are foundational in various fields such as physics, engineering, and computer science due to their ability to model infinite processes.

infinite series convergence

Infinite series convergence refers to whether the sum of an infinite series approaches a finite number as more terms are added. It's a vital concept in understanding the behavior of series in calculus.
To determine infinite series convergence, mathematicians employ various techniques:

• Comparison tests: Compare series terms with another series whose convergence is known.
• Ratio and root tests: Gauge how terms behave as \( n \) becomes very large, focusing on the idea of a limit.
• Integral test: Use integration to test the convergence by approximating the series as a function.
• P-Series test: Specifically applicable when series take the shape of a p-series.

Each method offers a different lens for assessing series behavior, providing a toolkit for tackling various infinite series scenarios. Ultimately, determining convergence is key for evaluating if an infinite sum approaches a particular value.