Problem 50
Question
Which of the series in Exercises \(17-56\) converge, and which diverge? Use any method, and give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{\tanh n}{n^{2}} $$
Step-by-Step Solution
Verified Answer
The series \( \sum_{n=1}^{\infty} \frac{\tanh n}{n^{2}} \) converges by the Comparison Test.
1Step 1: Identify the Series Type
The given series is \( \sum_{n=1}^{\infty} \frac{\tanh n}{n^{2}} \). We need to check if this series converges or diverges. The general term of the series is \( a_n = \frac{\tanh n}{n^{2}} \). This is a non-standard series, so we should compare it with a known convergent or divergent series.
2Step 2: Examine \( \tanh n \) Behavior
The hyperbolic tangent function, \( \tanh n \), is defined as \( \tanh n = \frac{e^n - e^{-n}}{e^n + e^{-n}} \). As \( n \to \infty \), \( \tanh n \to 1 \). Therefore, for large values of \( n \), \( \tanh n \) is close to 1.
3Step 3: Comparison with a p-series
Since \( \tanh n \) approaches 1, consider the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), which is a convergent p-series with \( p = 2 \). Because \( \tanh n \leq 1 \) for all \( n \) and \( \tanh n \to 1 \) as \( n \to \infty \), we can compare \( a_n = \frac{\tanh n}{n^2} \leq \frac{1}{n^2} \).
4Step 4: Use the Comparison Test
According to the Comparison Test, if \( 0 \leq a_n \leq b_n \) for all \( n \) and \( \sum b_n \) is convergent, then \( \sum a_n \) is also convergent. We established that \( a_n = \frac{\tanh n}{n^2} \leq \frac{1}{n^2} \), and \( \sum \frac{1}{n^2} \) is convergent. Thus, by the comparison test, \( \sum_{n=1}^{\infty} \frac{\tanh n}{n^2} \) converges.
Key Concepts
comparison testp-serieshyperbolic tangentnon-standard series
comparison test
The comparison test is a fundamental tool in determining whether a series converges or diverges.
To use the comparison test, you need to find a known series to compare against your given series. Generally, if you have a non-standard series, like in our case, you might look for a p-series or geometric series. The idea is to demonstrate that your series terms are smaller than or equal to the terms of a known convergent series, or greater than or equal to the terms of a known divergent series.
In this exercise, we compared our series with a p-series, showing that the series \(\sum_{n=1}^{\infty} \frac{\tanh n}{n^2}\) has terms \(a_n = \frac{\tanh n}{n^2}\) which are less than or equal to the terms of the convergent p-series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\). Hence, by the comparison test, if one series converges, so does the other.
To use the comparison test, you need to find a known series to compare against your given series. Generally, if you have a non-standard series, like in our case, you might look for a p-series or geometric series. The idea is to demonstrate that your series terms are smaller than or equal to the terms of a known convergent series, or greater than or equal to the terms of a known divergent series.
In this exercise, we compared our series with a p-series, showing that the series \(\sum_{n=1}^{\infty} \frac{\tanh n}{n^2}\) has terms \(a_n = \frac{\tanh n}{n^2}\) which are less than or equal to the terms of the convergent p-series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\). Hence, by the comparison test, if one series converges, so does the other.
p-series
A p-series is a specific kind of infinite series that is very useful in comparing with other series. It takes the form \(\sum_{n=1}^{\infty} \frac{1}{n^p}\), where \(p\) is a real number. Understanding the convergence of p-series is crucial because it often serves as a benchmark for comparison.
For a p-series, if \(p > 1\), the series converges. Conversely, if \(p \leq 1\), the series diverges. In the given exercise, we considered the p-series with \(p = 2\), which is convergent.
This insight was key because it provided us with a known benchmark for our comparison test. The term \(\frac{1}{n^2}\) provided a clear guideline for us to establish that our original series' terms were bounded, ensuring convergence.
For a p-series, if \(p > 1\), the series converges. Conversely, if \(p \leq 1\), the series diverges. In the given exercise, we considered the p-series with \(p = 2\), which is convergent.
This insight was key because it provided us with a known benchmark for our comparison test. The term \(\frac{1}{n^2}\) provided a clear guideline for us to establish that our original series' terms were bounded, ensuring convergence.
hyperbolic tangent
The hyperbolic tangent function, denoted as \(\tanh x\), is an interesting mathematical function often used in calculus and analysis. It is defined as \(\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}\). This function has properties that resemble the regular tangent function from trigonometry, but it relates to hyperbolic angles.
In the context of series convergence, knowing that \(\tanh n\) approaches 1 as \(n\) becomes very large was crucial. It implies that for long stretches of \(n\), \(\tanh n\) behaves consistently, staying between 0 and 1.
This consistent behavior allowed us to effectively use the comparison test, as \(\tanh n\) approaching 1 meant that \(\frac{\tanh n}{n^2}\) was always less than \(\frac{1}{n^2}\), permitting a straightforward comparison to a convergent series.
In the context of series convergence, knowing that \(\tanh n\) approaches 1 as \(n\) becomes very large was crucial. It implies that for long stretches of \(n\), \(\tanh n\) behaves consistently, staying between 0 and 1.
This consistent behavior allowed us to effectively use the comparison test, as \(\tanh n\) approaching 1 meant that \(\frac{\tanh n}{n^2}\) was always less than \(\frac{1}{n^2}\), permitting a straightforward comparison to a convergent series.
non-standard series
Non-standard series refer to infinite series that don't immediately fall into classic categories like geometric series, p-series, etc. They require a bit more creativity in analyzing their behavior and convergence.
Our exercise involved a non-standard series, \(\sum_{n=1}^{\infty} \frac{\tanh n}{n^2}\). Given its complexity due to the inclusion of the hyperbolic tangent function, this series didn't fit directly into the familiar molds.
To tackle such series, particularly for convergence or divergence, one often needs to use comparison tests with well-known series. This involves identifying the dominant behavior of the series' terms at infinity and finding a comparable series. The challenge and skill come in correctly simplifying or bounding the terms to frame the non-standard series in a recognizable manner, as shown with our p-series comparison approach.
Our exercise involved a non-standard series, \(\sum_{n=1}^{\infty} \frac{\tanh n}{n^2}\). Given its complexity due to the inclusion of the hyperbolic tangent function, this series didn't fit directly into the familiar molds.
To tackle such series, particularly for convergence or divergence, one often needs to use comparison tests with well-known series. This involves identifying the dominant behavior of the series' terms at infinity and finding a comparable series. The challenge and skill come in correctly simplifying or bounding the terms to frame the non-standard series in a recognizable manner, as shown with our p-series comparison approach.
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