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Chapter 9: Problem 42

Use the limit comparison test to determine whether the series converges or diverges.$$\sum \frac{n+1}{n^{2}+2}$$

Short Answer

Expert verified
The series diverges.

Step by step solution

01

Identify the Series

We are given the series \( \sum \frac{n+1}{n^{2}+2} \). Our task is to determine whether this series converges or diverges using the limit comparison test.
02

Select a Comparison Series

To apply the limit comparison test, we need to choose a known series to compare with. We choose the series \( b_n = \frac{1}{n} \) for our comparison because the highest degrees in the numerator and denominator of our original series suggest it behaves similarly to \( \frac{1}{n} \).
03

Apply the Limit Comparison Test

The limit comparison test states that for two series \( \sum a_n \) and \( \sum b_n \), if \( \lim_{{n \to \infty}} \frac{a_n}{b_n} = c \) where \( c \) is a positive finite number, then both series either converge or diverge together. Calculate the limit:\[ c = \lim_{{n \to \infty}} \frac{\frac{n+1}{n^{2}+2}}{\frac{1}{n}} = \lim_{{n \to \infty}} \frac{n(n+1)}{n^{2}+2} = \lim_{{n \to \infty}} \frac{n^2 + n}{n^2 + 2} \]Simplifying, this becomes:\[ c = \lim_{{n \to \infty}} \frac{1 + \frac{1}{n}}{1 + \frac{2}{n^2}} = \frac{1}{1} = 1 \]
04

Conclusion Based on Limit

Since \( c = 1 \), which is a positive finite number, the limit comparison test concludes that both \( \sum \frac{n+1}{n^{2}+2} \) and \( \sum \frac{1}{n} \) behave the same. The series \( \sum \frac{1}{n} \) is a harmonic series, which is known to diverge. Therefore, the original series \( \sum \frac{n+1}{n^{2}+2} \) also diverges.

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Key Concepts

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

Convergence
Convergence of a series is one of the foundational concepts in calculus and analysis. A series converges when the sum of its terms approaches a finite number as the number of terms increases indefinitely. When a series converges:
  • You can think of the terms settling into a sort of balance, leading to a specific sum.
  • This behavior is important in fields like physics and engineering, where stability and predictability are crucial.
To determine convergence, various tests like the Limit Comparison Test are used. This specific test compares your series to a known one whose behavior (convergence or divergence) is established. If the limit of the ratio of their terms approaches a positive finite constant, it indicates both series will behave the same way. The series \( \sum \frac{n+1}{n^{2}+2} \) is compared to the harmonic series in this manner, revealing insights into its behavior.
Divergence
Divergence, as opposed to convergence, describes the behavior where the sum of a series' terms does not settle into a specific finite number. Instead, it grows indefinitely. Some key characteristics of divergence include:
  • The series can overall increase without bound or oscillate without settling into a pattern.
  • Divergent series are often less useful directly in real-world applications but can be understood to extract valuable information.
The Limit Comparison Test stated that \( \lim_{{n \to \infty}} \frac{a_n}{b_n} = 1 \), a constant value, in our calculation. The outcome that \(\sum \frac{1}{n} \) diverges implies that \(\sum \frac{n+1}{n^{2}+2} \) also diverges, meaning it doesn鈥檛 sum to a finite value.
Harmonic Series
The harmonic series, expressed as \(\sum \frac{1}{n}\), is a classic example in the study of series, important due to its clear divergent nature. Though each term gets smaller as \(n\) increases, the series as a whole does not converge. A few notes about the harmonic series:
  • The series has historical importance, being studied since ancient Greek times and featuring prominently in mathematical analysis.
  • It helps illustrate the idea that a series can have diminishing terms yet still diverge.
This characteristic makes it an excellent candidate for comparison tests like the Limit Comparison Test. By showing that \(\lim_{{n \to \infty}} \frac{a_n}{b_n} = 1 \), where \(b_n = \frac{1}{n}\), it confirms that \(\sum \frac{n+1}{n^{2}+2}\) is divergent, just as the harmonic series is.

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Most popular questions from this chapter

Determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{(n-1) !}{n^{2}}$$

What values of \(a\) does the series converge? $$\sum_{n=1}^{\infty}\left(\frac{2}{n}\right)^{a}$$

Decide if the statements are true or false. Give an explanation for your answer.$$\text { If } \sum a_{n} \text { converges, then } \lim _{n \rightarrow \infty}\left|a_{n+1}\right| /\left|a_{n}\right| \neq 1$$.

Determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{n(n+1)}{\sqrt{n^{3}+2 n^{2}}}$$

The sequence \(s_{n}\) is increasing, the sequence \(t_{n}\) converges, and \(s_{n} \leq t_{n}\) for all \(n .\) Show that \(s_{n}\) converges.
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