Problem 23
Question
Which of the series in Exercises \(17-56\) converge, and which diverge? Use any method, and give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{10 n+1}{n(n+1)(n+2)} $$
Step-by-Step Solution
Verified Answer
The series diverges.
1Step 1: Simplify the General Term
The series given is \( \sum_{n=1}^{\infty} \frac{10n+1}{n(n+1)(n+2)} \). We need to find a simpler expression for the general term \( a_n = \frac{10n+1}{n(n+1)(n+2)} \) that represents this series.To do this, we'll use partial fraction decomposition to express the term as a sum of simpler fractions.Since \( a_n \) has the form of \( \frac{1}{n(n+1)(n+2)} \), let's set up:\[\frac{10n+1}{n(n+1)(n+2)} = \frac{A}{n} + \frac{B}{n+1} + \frac{C}{n+2}\]Multiplying through by \( n(n+1)(n+2) \) gives:\[10n + 1 = A(n+1)(n+2) + Bn(n+2) + Cn(n+1)\]
2Step 2: Solve for Partial Fractions Coefficients
To find \( A \), \( B \), and \( C \), expand and equate coefficients.1. Expand: \( A(n^2 + 3n + 2) + B(n^2 + 2n) + C(n^2 + n) \)2. Collect terms: \((A + B + C)n^2 + (3A + 2B + C)n + 2A)\)3. Comparing with \( 10n + 1 \), we get: - \( A + B + C = 0 \) - \( 3A + 2B + C = 10 \) - \( 2A = 1 \)Solving these equations, we find: \( A = \frac{1}{2} \) \( B = -2 \) \( C = \frac{3}{2} \)
3Step 3: Rewrite the Series with Partial Fractions
Replace \( \frac{10n+1}{n(n+1)(n+2)} \) using \( A \), \( B \), and \( C \):\[\frac{10n+1}{n(n+1)(n+2)} = \frac{1}{2n} - \frac{2}{n+1} + \frac{3}{2(n+2)}\]Rewriting the series:\[\sum_{n=1}^{\infty} \left(\frac{1}{2n} - \frac{2}{n+1} + \frac{3}{2(n+2)}\right)\]
4Step 4: Determine If the Series Converges
To determine convergence, recognize that each of these terms results in a telescoping series:Re-expressing each series:- \( \frac{1}{2n} \) terms do not telescope and hence diverge.- The combination \(-\frac{2}{n+1} + \frac{3}{2(n+2)}\) suggests telescoping possibly when summed appropriately, but the critical \( \frac{1}{2n} \) ensures divergence.Since the term \( \frac{1}{2n} \) behaves like the harmonic series which diverges, the series overall diverges.
Key Concepts
Partial fraction decompositionTelescoping seriesHarmonic seriesDivergence
Partial fraction decomposition
Partial fraction decomposition is a technique used to simplify complex rational expressions. It involves breaking down a complex fraction into a sum of simpler fractions. This simplification makes it easier to analyze the behavior of series and integrals.
In this problem, the term \( \frac{10n+1}{n(n+1)(n+2)} \) is split into simpler partial fractions such that:
In this problem, the term \( \frac{10n+1}{n(n+1)(n+2)} \) is split into simpler partial fractions such that:
- \( \frac{A}{n} \)
- \( \frac{B}{n+1} \)
- \( \frac{C}{n+2} \)
Telescoping series
A telescoping series is a series whose terms cancel each other out over successive additions. This makes it possible to find the sum of the series more directly.
In a telescoping series, consecutive terms are structured such that most terms are eliminated when summed. Essentially, only a few terms from the beginning and the end of the series contribute to the sum. This cancellation helps precisely determine convergence or divergence.
In the exercise, the decomposition into partial fractions suggests a telescoping nature, as terms such as \( -\frac{2}{n+1} \) and \( \frac{3}{2(n+2)} \) can potentially cancel each other in a specific pattern. While these might simplify, it is crucial to recognize that some terms, particularly those behaving like harmonic series terms, may still lead the overall series to diverge.
In a telescoping series, consecutive terms are structured such that most terms are eliminated when summed. Essentially, only a few terms from the beginning and the end of the series contribute to the sum. This cancellation helps precisely determine convergence or divergence.
In the exercise, the decomposition into partial fractions suggests a telescoping nature, as terms such as \( -\frac{2}{n+1} \) and \( \frac{3}{2(n+2)} \) can potentially cancel each other in a specific pattern. While these might simplify, it is crucial to recognize that some terms, particularly those behaving like harmonic series terms, may still lead the overall series to diverge.
Harmonic series
The harmonic series is a well-known series in mathematics defined as \( \sum_{n=1}^{\infty} \frac{1}{n} \).
Despite the terms approaching zero, the harmonic series diverges. This is because the sum grows without bound as more terms are added.
In the given problem, the term \( \frac{1}{2n} \) behaves similarly to the harmonic series. Even when combined with other terms in the decomposition, this critical component implies that the series does not converge. Recognizing parts of a series that behave like a harmonic series is essential to understanding the overall behavior of the series.
Despite the terms approaching zero, the harmonic series diverges. This is because the sum grows without bound as more terms are added.
In the given problem, the term \( \frac{1}{2n} \) behaves similarly to the harmonic series. Even when combined with other terms in the decomposition, this critical component implies that the series does not converge. Recognizing parts of a series that behave like a harmonic series is essential to understanding the overall behavior of the series.
Divergence
Divergence occurs when the terms of a series do not sum to a finite limit as more terms are added. In other words, the series grows without bound.
For the given series \( \sum_{n=1}^{\infty} \frac{10n+1}{n(n+1)(n+2)} \), the partial fractions simplify complex terms, and initially suggest possible telescoping. However, the component \( \frac{1}{2n} \) signifies divergence due to its similarity to the harmonic series.
When addressing convergence or divergence, it is critical to examine each term's influence on the series carefully. Here, the divergence of the harmonic-like term overshadowed any potential benefits from telescoping, leading to the conclusion that the original series diverges.
For the given series \( \sum_{n=1}^{\infty} \frac{10n+1}{n(n+1)(n+2)} \), the partial fractions simplify complex terms, and initially suggest possible telescoping. However, the component \( \frac{1}{2n} \) signifies divergence due to its similarity to the harmonic series.
When addressing convergence or divergence, it is critical to examine each term's influence on the series carefully. Here, the divergence of the harmonic-like term overshadowed any potential benefits from telescoping, leading to the conclusion that the original series diverges.
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