Problem 31 - 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} \frac{1}{1+\ln n}\end{equation}

Step-by-Step Solution

Answer

The series diverges because it behaves like the divergent harmonic series.

Step 1: Determine the Form of the Series

The given series is \( \sum_{n=1}^{\infty} \frac{1}{1+\ln n} \). This means that for each term in the series, the denominator is \(1 + \ln n\), where \(n\) is a positive integer. As \(n\) increases, \(\ln n\) increases, making \(\frac{1}{1+\ln n}\) decrease.

Step 2: Compare with a Known Series

Consider the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is known to diverge. The given series can be compared to this, particularly by noticing that \( \ln n < n \) for all \(n > 1\). This implies that \( \frac{1}{1+\ln n} > \frac{1}{n} \) for sufficiently large \(n\).

Step 3: Use the Limit Comparison Test

To determine convergence or divergence, apply the Limit Comparison Test. Compare our series with the harmonic series: \( b_n = \frac{1}{n} \). Calculate \( \lim_{n \to \infty} \frac{\frac{1}{1+\ln n}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n}{1+\ln n} \).

Step 4: Evaluate the Limit

Compute the limit \( \lim_{n \to \infty} \frac{n}{1+\ln n} \). As \(n\) approaches infinity, \(\ln n\) grows slower than \(n\), so the expression simplifies to \( \infty \). This suggests the original series diverges, as the denominator grows much slower than the numerator.

Step 5: Conclusion on Convergence

Since \( \lim_{n \to \infty} \frac{n}{1+\ln n} = \infty \), and knowing that the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges, the Limit Comparison Test implies that the series \( \sum_{n=1}^{\infty} \frac{1}{1+\ln n} \) also diverges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a useful tool when working with series to determine their convergence or divergence. This test is most handy when you have a series that looks similar to another series with a known behavior. To apply it, you follow these steps:

• Select a comparison series \(b_n\) that shares similar characteristics with your series \(a_n\).
• Compute the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If the limit is a positive finite number, say \(c\), then both \( \sum a_n \) and \( \sum b_n \) either converge or diverge together.

This test is particularly potent when \(a_n\) and \(b_n\) are both positive and \(b_n\) is well understood, like the harmonic series \( \sum \frac{1}{n} \). In our original problem, the series \( \sum_{n=1}^{\infty} \frac{1}{1+\ln n} \) is compared to its harmonic cousin. The result shows the limit as \( \infty \), aligning with the divergence of the harmonic series.

Divergent Series

A series is considered divergent if the sum of its terms grows without bound as more terms are added. Divergence indicates that adding terms to the series does not bring it closer to a finite value. An essential example of a divergent series is the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \).

• The harmonic series diverges, meaning its terms approach infinity as you sum from term one to infinity.
• Divergence contrasts with convergence, where a series approaches a specific value.

For the problem we are dealing with,\( \sum_{n=1}^{\infty} \frac{1}{1+\ln n} \) is shown to diverge. This conclusion comes from comparing it to the divergent harmonic series using the Limit Comparison Test we just discussed. Identifying a series as divergent is crucial in mathematics as it defines the behavior and applicability of certain calculations and models.

Harmonic Series

The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is one of the simplest yet most significant examples of a divergent series in mathematics. Despite its simple form, it has intriguing properties:

• The harmonic series grows logarithmically, meaning its partial sums increase slowly, yet steadily, towards infinity.
• Every added term in the harmonic series includes smaller contributions than the previous, reinforcing that no finite limit is ever reached.

An understanding of the harmonic series is valuable when analyzing other series. In our case, its divergence acted as a reference for the series \( \sum_{n=1}^{\infty} \frac{1}{1+\ln n} \). By comparing them, we've established that if the more complex series diverges slower, or at the same rate, as the harmonic series, it too diverges. This demonstration solidifies the usefulness of the harmonic series as a benchmark in convergence tests.