Problem 46 - Thomas Calculus

Question

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \tan \frac{1}{n}\end{equation}

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \tan \frac{1}{n} \) diverges.

Step 1: Analyze the Series

To determine if the series \( \sum_{n=1}^{\infty} \tan \frac{1}{n} \) converges or diverges, let's examine the general term \( a_n = \tan \frac{1}{n} \). For large \( n \), \( \frac{1}{n} \) approaches 0, and we can use the approximation \( \tan x \approx x \) for small values of \( x \). Thus, \( \tan \frac{1}{n} \approx \frac{1}{n} \).

Step 2: Compare with Harmonic Series

Given that for large \( n \), \( \tan \frac{1}{n} \approx \frac{1}{n} \), the series \( \sum_{n=1}^{\infty} \tan \frac{1}{n} \) behaves similarly to the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \). The harmonic series is known to diverge. Therefore, the behavior of \( \tan \frac{1}{n} \) suggests that our series likely diverges too.

Step 3: Use Direct Comparison Test

The Direct Comparison Test can confirm divergence. If \( a_n = \tan \frac{1}{n} \approx \frac{1}{n} \), and given that \( \frac{1}{n} \) diverges, we should compare \( \tan \frac{1}{n} \) directly to \( \frac{1}{n} \). Note that \( \tan \frac{1}{n} > \frac{1}{n}^2 \) for large \( n \), and \( \sum \frac{1}{n}^2 \) converges (p-series with \( p = 2 \)), but this does not imply convergence for \( \tan \frac{1}{n} \). Hence, without a smaller convergent comparison, and since it's approximately \( \frac{1}{n} \), the series \( \sum \tan \frac{1}{n} \) diverges.

Key Concepts

Harmonic Series

In mathematics, the harmonic series is a well-known example of a divergent series. It is represented as \[\sum_{n=1}^{\infty} \frac{1}{n}\]This series is simple to understand: it is just the sum of the reciprocals of all natural numbers. Despite its straightforward appearance, it might be surprising to learn that it diverges. This means that if you keep adding its terms, the sum will grow without bound.

The divergence of the harmonic series can be demonstrated through various methods, such as the integral test or by grouping terms in a specific way. A key takeaway is that even though each term gets smaller and smaller, they don't decrease fast enough for the series to settle on a finite value. Understanding the harmonic series is crucial because it serves as a reference for evaluating other series, especially when comparing their convergence behavior.

Direct Comparison Test

The Direct Comparison Test is a helpful tool to determine the convergence or divergence of a series by comparing it to another series with known behavior. Here's how it works:

• Compare the given series to a well-known series where you already know whether it converges or diverges.
• If you can find a series with terms larger than the original series that converges, the original series converges too.
• If you find a series with terms smaller than the original series that diverges, the original series must also diverge.

A classic example involves comparing a complex series against the harmonic series. Given the series \[\sum_{n=1}^{\infty} \tan \frac{1}{n}\]we approximate \( \tan \frac{1}{n} \approx \frac{1}{n} \)for large \( n \), suggesting a similarity to the harmonic series. Since the harmonic series diverges, the Direct Comparison Test helps reinforce that \( \sum \tan \frac{1}{n} \)diverges as well.

p-Series

A p-series is another essential series type used to analyze series in calculus. It is expressed as:\[\sum_{n=1}^{\infty} \frac{1}{n^p}\]where \( p \) is a positive constant. Understanding the convergence of a p-series is simplified by the value of \( p \):

• If \( p > 1 \), the series converges.
• If \( p \leq 1 \), the series diverges.

For instance, the series \( \sum \frac{1}{n^2} \)converges because \( p = 2 > 1 \). This contrasts with the harmonic series where \( p = 1 \), leading it to diverge. This basic rule can be applied to explore whether more complex series converge or diverge by comparing their structure to a p-series. Thus, it offers a fundamental understanding of infinite series behavior in calculus.