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Chapter 9: Problem 13

Does the series converge or diverge? $$\sum_{n=0}^{\infty} \frac{3}{n+2}$$

Short Answer

Expert verified
The series diverges by comparison to the harmonic series.

Step by step solution

01

Identify the Series

We are given the series \( \sum_{n=0}^{\infty} \frac{3}{n+2} \). This is an infinite series where the general term is \( a_n = \frac{3}{n+2} \).
02

Consider the Divergence Test

The divergence test states that if \( \lim_{n \to \infty} a_n eq 0 \), then the series diverges. We need to find \( \lim_{n \to \infty} \frac{3}{n+2} \).
03

Calculate the Limit of the General Term

Evaluate the limit: \( \lim_{n \to \infty} \frac{3}{n+2} = \lim_{n \to \infty} \frac{3}{n} = 0 \). However, since the terms decrease to 0, this is necessary for convergence but not sufficient.
04

Comparison with Harmonic Series

The general term \( \frac{3}{n+2} \) behaves similarly to the harmonic series term \( \frac{1}{n} \), which is known to diverge. Since \( \frac{3}{n+2} > \frac{3}{n+3} \) for all \( n \geq 1 \), and the terms of the harmonic series diverge, \( \sum \frac{3}{n+2} \) also diverges.
05

Conclusion

Since the terms \( \frac{3}{n+2} \) decrease to zero but are greater than that of a divergent harmonic series, the series \( \sum_{n=0}^{\infty} \frac{3}{n+2} \) also diverges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convergence Tests
In the realm of infinite series, understanding whether a series converges or diverges is crucial. Convergence tests are a set of methods or criteria used to determine the behavior of a series. A series converges if its sequence of partial sums approaches a finite limit. Otherwise, it diverges.

Some common convergence tests include:
  • Divergence Test: If the limit of the general term of a series as it approaches infinity is not zero, then the series diverges.
  • Comparison Test: Compares the series in question to a known series to determine convergence or divergence.
  • Ratio Test: Determines convergence by examining the ratio of successive terms.
  • Integral Test: Uses integrals to test convergence of series by comparing to function behavior.
For example, in the series provided, we used the Comparison Test against the harmonic series, known for its divergence. Such tests are foundational tools in calculus to analyze series behavior.
Harmonic Series
The harmonic series is a common example used to explain concepts of series' convergence and divergence. It's defined as \( \sum_{n=1}^{\infty} \frac{1}{n} \). This series is one of the most well-known examples of a series that diverges, despite its terms decreasing and approaching zero.

Characteristics of the harmonic series include:
  • Its terms form a sequence that decreases towards zero.
  • Despite the decreasing terms, the series does not approach a finite sum.
  • Comparing other series to the harmonic series helps in determining divergence for series with similar terms.
In our exercise, \( \frac{3}{n+2} \) was compared to the harmonic series to determine its divergence. Since \( \frac{3}{n+2} \) mimics \( \frac{3}{n+3} \), and given that \( \frac{1}{n} \) diverges, the series in question does not converge either.
Divergence Test
The divergence test is the simplest tool in assessing whether a series diverges. Although it does not confirm convergence, it provides a straightforward way to identify divergence. If the general term of a series does not approach zero, the series must diverge.

Key aspects of the divergence test include:
  • It is used as an initial check for series analysis.
  • If \( \lim_{n \to \infty} a_n eq 0 \, \) the series definitely diverges.
  • This test alone cannot prove convergence even if the limit is zero.
In the provided problem, the limit \( \lim_{n \to \infty} \frac{3}{n+2} = 0 \) was calculated. While not sufficient for convergence proof, it indicated the need for further tests like the comparison with the harmonic series, resulting in identifying divergence.

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Most popular questions from this chapter

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