Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{10 n+1} $$

Step-by-Step Solution

Answer

The series is divergent.

Step 1: Analyze the Form of the Series

The given series is \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{10n+1} \). This is an alternating series, as indicated by the factor \((-1)^{n+1}\). Alternating series have the general form \(\sum (-1)^n a_n\), where \(a_n > 0\). In our case, \(a_n = \frac{n}{10n+1}\).

Step 2: Check the Condition for the Alternating Series Test

The alternating series test states that an alternating series \(\sum (-1)^n a_n\) converges if the sequence \(a_n\) is positive, decreasing, and approaches zero as \(n\) approaches infinity. Here, \(a_n = \frac{n}{10n+1}\). Check if \(a_n\) is decreasing and approaches zero.

Step 3: Show \( a_n \) Decreases to Zero

Calculate \(\lim_{n \to \infty} \frac{n}{10n+1}\). This can be done by dividing the numerator and denominator by \(n\), yielding \(\lim_{n \to \infty} \frac{1}{10 + \frac{1}{n}} = \frac{1}{10}\). Therefore, \(a_n\) does not approach zero. This means the alternating series test fails, implying that the series is divergent.

Step 4: Check Absolute Convergence

For absolute convergence, consider the series \(\sum_{n=1}^{\infty} \left| \frac{n}{10n+1} \right| = \sum_{n=1}^{\infty} \frac{n}{10n+1}\). This is a series of positive terms, and we suspect it diverges because \(\frac{n}{10n+1} \approx \frac{1}{10}\) for large \(n\), suggesting the terms do not go to zero.

Step 5: Use a Comparison Test

Compare \(\sum \frac{n}{10n+1}\) to \(\sum \frac{1}{10}\). Since \(\frac{n}{10n+1} \sim \frac{1}{10}\) for large \(n\), and \(\sum \frac{1}{10}\) is a divergent harmonic series, \(\sum \frac{n}{10n+1}\) diverges by the comparison test. Therefore, the original series is neither absolutely convergent nor conditionally convergent.

Key Concepts

Absolute Convergence

When we talk about absolute convergence, we are interested in whether the series of absolute values converges. This means we take every term in the series, apply absolute values, and check if the series still converges.
For our given series, we looked at:

• he series without the alternating sign,
• which is \ \( \sum_{n=1}^{\infty} \frac{n}{10n+1} \ \).

We suspected that this series diverges because the terms \( \frac{n}{10n+1} \) behave like \( \frac{1}{10} \) for large \( n \). This rough approximation suggests the terms do not decrease to zero, which is necessary for convergence. Thus, the series fails to be absolutely convergent since the absolute values do not form a convergent series.

Conditional Convergence

Conditional convergence refers to when a series converges, but does not converge absolutely. In this case, merely looking at the signs and values combined, without taking absolute values, reveals convergence. For the series in question, \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{10n+1} \), we're checking if removing absolute values still results in convergence.
However, as analyzed using the Alternating Series Test, this series does not satisfy the condition that the sequence \( a_n = \frac{n}{10n+1} \) decreases and approaches zero. Specifically, \( a_n \) does not go to zero, thus the series is not conditionally convergent. Instead, it simply diverges, making an interesting point about when alternating series can and can't converge conditionally.

Comparison Test

The Comparison Test helps us determine the convergence or divergence of a series by comparing it to another series that is easily understood.
Here, we use the non-alternating series \( \sum_{n=1}^{\infty} \frac{n}{10n+1} \) and compare it to the series \( \sum \frac{1}{10} \).

• If a series \( b_n \) is known to diverge and we can show that \( a_n \) is greater than or roughly equal to \( b_n \), the comparison test tells us \( a_n \) must also diverge.
• In this case, \( \frac{n}{10n+1} \) behaves like \( \frac{1}{10} \) when \( n \) is large, due to it simplifying approximately to \( \frac{1}{10} \) leading it to diverge as well.

This comparison confirmed the divergence of \( \sum \frac{n}{10n+1} \), allowing us to conclude that the original series is not absolutely or conditionally convergent, but rather divergent.

Divergence in Series

Divergence in series indicates that the sum does not stabilize or approach a finite limit as more terms are added. Several common signs point to a series diverging.
For the given series \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{10n+1} \), a couple factors led to identifying its divergence:

• The Alternating Series Test revealed that the terms \( a_n = \frac{n}{10n+1} \), do not approach zero.
• The approximate and comparison reasoning showed they behave like the harmonic series, which inherently diverges.

This means as we sum endlessly, the total does not settle down to a specific value. Instead, it continues growing, revealing the nature of divergence in this series.