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Magazine Chapter 11: Problem 24
Use the limit comparison test to determine whether the series converges or diverges. $$ \sum_{n=1}^{x} \frac{1}{\sqrt{n(n+1)(n+2)}} $$
Short Answer
Expert verified
The series converges by the limit comparison test.
Step by step solution
01
Identify a Comparable Series
We need to compare the given series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n(n+1)(n+2)}} \) with a known convergent or divergent series. Notice that the expression inside the square root \( \sqrt{n(n+1)(n+2)} \) can be approximated to \( n^{3/2} \) for large \( n \). Thus, a comparable series is \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \), which is a p-series with \( p = 3/2 > 1 \), known to converge.
02
Apply the Limit Comparison Test
The limit comparison test involves calculating the limit: \[ \lim_{n \to \infty} \frac{a_n}{b_n} \] where \( a_n = \frac{1}{\sqrt{n(n+1)(n+2)}} \) and \( b_n = \frac{1}{n^{3/2}} \). Evaluate this limit: \[ \lim_{n \to \infty} \frac{1/\sqrt{n(n+1)(n+2)}}{1/n^{3/2}} = \lim_{n \to \infty} \frac{n^{3/2}}{\sqrt{n(n+1)(n+2)}} \].
03
Simplify the Limit Expression
To simplify the limit expression \( \lim_{n \to \infty} \frac{n^{3/2}}{\sqrt{n(n+1)(n+2)}} \), factor out \( n^3 \) under the square root of the denominator:\[\lim_{n \to \infty} \frac{n^{3/2}}{\sqrt{n^3 (1+1/n)(1+2/n)}} = \lim_{n \to \infty} \frac{n^{3/2}}{n^{3/2}\sqrt{(1+1/n)(1+2/n)}}.\]
04
Evaluate the Limit
The \( n^{3/2} \) terms in the numerator and denominator cancel each other, leaving: \[\lim_{n \to \infty} \frac{1}{\sqrt{(1+1/n)(1+2/n)}} = \lim_{n \to \infty} \frac{1}{\sqrt{1+1/n} \cdot \sqrt{1+2/n}}.\]As \( n \to \infty \), \( 1/n \) and \( 2/n \) approach 0, thus the limit is:\[\frac{1}{\sqrt{1} \cdot \sqrt{1}} = 1.\]
05
Conclude Using Limit Comparison
Since the limit is finite and positive (1 in this case), the original series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n(n+1)(n+2)}} \) and the comparison series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) exhibit the same behavior. Since \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) converges, so does the original series.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convergence of Series
Series convergence is a fundamental topic in mathematics. Here, we focus on determining whether an infinite series converges or diverges. A series converges when its sum approaches a fixed number as more terms are added.
This involves investigating the behavior of the terms as they go towards infinity.
There are several methods to determine convergence:
This involves investigating the behavior of the terms as they go towards infinity.
There are several methods to determine convergence:
- Comparison Tests: Comparing a series to a known convergent or divergent series.
- Integral Tests: Relating the series to a corresponding integral.
- Ratio and Root Tests: Using the limits of ratios or roots of terms.
P-Series Convergence
Understanding the convergence of a p-series is key in solving many series convergence problems.
A p-series takes the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), with \( p \) being a positive constant. The convergence of a p-series strongly depends on the value of \( p \):
This is a p-series with \( p = \frac{3}{2} \), which means it converges.
The knowledge of p-series behavior is crucial to applying the Limit Comparison Test effectively, as it provides us with a known series to compare against for determining convergence.
A p-series takes the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), with \( p \) being a positive constant. The convergence of a p-series strongly depends on the value of \( p \):
- If \( p > 1 \), the series converges.
- If \( p \leq 1 \), the series diverges.
This is a p-series with \( p = \frac{3}{2} \), which means it converges.
The knowledge of p-series behavior is crucial to applying the Limit Comparison Test effectively, as it provides us with a known series to compare against for determining convergence.
Mathematical Limits
Limits are a vital concept when working with infinite series. They help us understand how the terms of a series behave as they proceed to infinity.
In mathematical analysis, limits are used to:
By concluding that the limit is finite and positive, typically 1, both series share the same convergence behavior.
In mathematical analysis, limits are used to:
- Evaluate the behavior or sum of an infinite series.
- Simplify expressions for terms in a sequence or series.
By concluding that the limit is finite and positive, typically 1, both series share the same convergence behavior.
Series Comparison
The Limit Comparison Test is a specific method for evaluating the convergence of a series by comparison.
This test involves:
In our case, the series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) serves as an easy benchmark due to its well-known convergence.
By calculating the limit and finding it is 1, it confirms that the original series converges.
This makes series comparison a powerful strategy in dealing with infinite series observations.
This test involves:
- Finding a comparable series with known convergence properties.
- Calculating the limit of the ratio of their terms as \( n \to \infty \).
- Deciding the original series behavior based on the limit.
In our case, the series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) serves as an easy benchmark due to its well-known convergence.
By calculating the limit and finding it is 1, it confirms that the original series converges.
This makes series comparison a powerful strategy in dealing with infinite series observations.
