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Magazine Chapter 11: Problem 36
Determine the values of \(p\) for which the given series converges. \(\sum_{n=1}^{\infty} \frac{n}{\left(n^{2}+1\right)^{p}}\)
Short Answer
Expert verified
The series converges for \( p > 1 \).
Step by step solution
01
Express the Series Using a Test
Consider the series \( \sum_{n=1}^{\infty} \frac{n}{\left(n^{2}+1\right)^{p}} \). To determine convergence, we will use the Limit Comparison Test or, more relevantly for this form, the Integral Test.
02
Compare to a Similar Series
We note that for large \( n \), \( n^2 + 1 \approx n^2 \). Hence, the series behaves like \( \sum \frac{n}{(n^2)^p} = \sum \frac{n}{n^{2p}} = \sum \frac{1}{n^{2p-1}} \) for large \( n \). This is analogous to a p-series \( \sum \frac{1}{n^q} \).
03
Determine Convergence of the p-series
From the p-series test, we know \( \sum \frac{1}{n^q} \) converges if \( q > 1 \). Applying this to our series, we have \( q = 2p - 1 \). Therefore, \( 2p - 1 > 1 \) must be true to ensure convergence.
04
Solve the Inequality
Solving the inequality \( 2p - 1 > 1 \), we get: \[ 2p > 2 \] \[ p > 1 \].Thus, the series converges for \( p > 1 \).
05
Verify with the Integral Test
For further verification, consider the integral \( \int \frac{x}{(x^2+1)^p} \, dx \). Convergence of this integral for \( p > 1 \) supports the result obtained with the p-series test.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit Comparison Test
The Limit Comparison Test is a powerful tool for determining the convergence or divergence of a series. It is particularly useful when the given series resembles a simpler series whose behavior is known. To apply this test, compare the series in question to a second series \( \sum b_n \) whose convergence is known. If the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \) is a positive finite number, both series either converge or diverge together.
- Choose a comparison series, ideally one that resembles the original series as \( n \) becomes very large.
- Calculate the limit of the ratio of the terms of the two series.
- If the limit is positive and finite, the behavior (convergent/divergent) of both series is the same.
p-series Test
The p-series test is a straightforward method to analyze series of the form \( \sum \frac{1}{n^q} \). Such a series is known to converge when \( q > 1 \), and diverge when \( q \leq 1 \). This fundamental test is critical because many series can be manipulated or approximated to resemble a p-series.
- Identify if the series can be written or compared to \( \sum \frac{1}{n^q} \).
- Evaluate the exponent \( q \).
- Apply the test: convergence if \( q > 1 \), otherwise divergence.
Integral Test
The Integral Test is another valuable tool for determining series convergence. This test asserts that if \( f(x) \) is a positive, continuous, decreasing function on \( [1, \infty) \), then the series \( \sum f(n) \) converges if and only if the integral \( \int_1^{\infty} f(x) \, dx \) converges.
- Check that \( f(x) \) is positive, continuous, and decreasing.
- Compute the integral from 1 to infinity.
- Convergence of the integral implies convergence of the series, and vice versa.
Inequalities in Convergence Analysis
Inequalities play an essential role in determining series convergence, particularly when linked to tests like the p-series and integral tests. The exercise involves solving inequalities to establish the convergence criteria. For example, \( 2p - 1 > 1 \) leads to \( 2p > 2 \), simplifying to \( p > 1 \).
- Inequalities help us isolate the variables that affect convergence.
- Simplify complex expressions to find clear conditions.
- Often involve straightforward algebra to solve.
