Problem 53 - Essential Calculus

Question

In Exercises \(53-68,\) determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n+10}{10 n+1} $$

Step-by-Step Solution

Answer

The series \(\sum_{n=1}^{\infty} \frac{n+10}{10n+1}\) diverges.

Step 1: Identify the Series Type

Identify the general term of the series, which in this case is \(a_n = \frac{n+10}{10n+1}\). The series involves a rational expression, which suggests that we could apply the comparison test.

Step 2: Apply The Limit Comparison Test

Choose a second series that we can compare with our original series. A good choice is \(b_n = \frac{n}{n}=1\), as it is a simple series to compare with and we can determine its convergence easily. Apply the Limit Comparison Test

Step 3: Calculate The Limit

Calculate the limit as n approaches infinity of \(\frac{a_n}{b_n}\), which in this case is \(\frac{\frac{n+10}{10n+1}}{1}\). By direct simplification, we have \(\lim_{n\to\infty} \frac{n+10}{10n+1} = \lim_{n \to \infty} \frac{n}{10n} = \frac{1}{10}\)

Step 4: Interpret the result

Since the limit is a finite positive number, according to the limit comparison test, the original series has the same convergence behavior as the series we chose for comparison, which in this case is \(\sum_{n=1}^{\infty} 1\). This series is a harmonic series that diverges.

Key Concepts

Limit Comparison Test

When faced with complex series, the limit comparison test is a powerful tool to determine convergence or divergence. It involves comparing the series in question with another known series by calculating the limit of the ratio of their terms as the index approaches infinity. If this limit is a finite positive number, then both series will either converge or diverge together.
The steps for applying this test include:

• Identifying the general term of the series.
• Choosing a comparison series with known behavior.
• Calculating the limit of the ratio of the series terms.
• Interpreting the limit according to the convergence behavior of the comparison series.

This technique simplifies the process of analyzing complex series through a more familiar benchmark.

Harmonic Series

A harmonic series is a particular example of an infinite series that is especially significant due to its divergence.
The harmonic series has the form:
\[\sum_{n=1}^{\infty} \frac{1}{n}\]
This series increases without bound, thus diverging. The significance in the context of the limit comparison test lies in its role as a benchmark for comparing series. If another series can be shown to behave like the harmonic series in this regard, it can also be deemed divergent.
Understanding the harmonic series and its divergence provides a foundational reference when analyzing the convergence or divergence of other series.

Rational Expressions

Rational expressions are ratios of polynomial expressions. They often appear in the terms of series, such as \[\frac{n+10}{10n+1},\]the term used in this specific task.
Polynomials, due to their straightforward characteristics, allow us to simplify and analyze series. In such cases, simplification often involves dividing the leading coefficients, which can provide insights into the series' behavior as \(n\) approaches infinity.
This simplification is paramount when applying tools like the limit comparison test, as it enables finding the comparison series and calculating the necessary limits.

Infinite Series

Infinite series involve the summation of an infinite sequence of terms. The goal in studying infinite series is to determine their convergence or divergence.
Convergence means the series approaches a finite value as more terms are added. Divergence implies there is no finite sum.
Analyzing infinite series encompasses several tests and methods, such as the limit comparison test, and understanding basic series forms like the geometric or harmonic series.
Mastering the concept of infinite series and recognizing patterns becomes crucial in advanced calculus and mathematical analysis, extending its applications in various fields such as engineering, physics, and beyond.