Problem 48 - Thomas Calculus

Question

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ 1+\frac{1}{4}-\frac{1}{9}-\frac{1}{16}+\frac{1}{25}+\frac{1}{36}-\frac{1}{49}-\frac{1}{64}+\cdots $$

Step-by-Step Solution

Answer

The series converges absolutely because the absolute value series is a convergent p-series.

Step 1: Understanding the Series

The given series is:\[ 1+\frac{1}{4}-\frac{1}{9}-\frac{1}{16}+\frac{1}{25}+\frac{1}{36}-\frac{1}{49}-\frac{1}{64}+\cdots \]This series includes alternating positive and negative terms that have a specific pattern. We need to determine if this series converges absolutely, converges conditionally, or diverges.

Step 2: Examine the Pattern of Terms

Observe the series and recognize that the numerator alternates between two positive terms followed by two negative terms. Additionally, each term is of the form \( \frac{1}{n^2} \), where \( n = 1, 2, 3, \ldots \).

Step 3: Analyze the Absolute Series

To determine absolute convergence, consider the series made up of the absolute values of the terms:\[ 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots \]This is a p-series with \( p = 2 \), which is known to converge since \( p > 1 \).

Step 4: Determine Absolute Convergence

Since the series of the absolute values of the terms converges (as shown by the p-series comparison), the original series also converges absolutely.

Key Concepts

Absolute Convergence

When we speak of absolute convergence in the context of infinite series, it means that the series will still converge even if we take the absolute values of its terms. In simpler terms, if \( \sum |a_n| \) converges, then the series \( \sum a_n \) is said to absolutely converge. This concept is incredibly useful because it provides a stronger condition than mere convergence.

In our specific example, after taking the absolute value, the series becomes \[ 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots \] which is a well-known type of series called a p-series. With \( p = 2 \), this series converges, as we will explore more in the p-series section.

This conclusion tells us that the original series converges absolutely. Absolute convergence implies conditional convergence, so if a series converges absolutely, it definitely converges in the conventional sense as well.

P-Series

The p-series is a critical concept in understanding series convergence. A p-series is defined as a series of the form \[ \sum \frac{1}{n^p} \] where \( n \) is a positive integer, and \( p \) is a real number. Its convergence largely depends on the value of \( p \).

**Key Facts about P-Series:**

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

In our exercise, the absolute value series is a p-series with \( p = 2 \). According to the rules above, since \( p = 2 \) is greater than 1, this series converges. Therefore, by checking the p-series condition, we've also verified that the series of absolute values is convergent, solidifying our earlier conclusion of absolute convergence for the original series.

Conditional Convergence

Conditional convergence occurs when a series converges, but does not converge absolutely. This concept is a bit more nuanced and requires us to focus on the original series' alternating terms.

In some series, like the alternating harmonic series, conditional convergence happens because the terms are large enough to keep the absolute series from converging, yet the alternating nature allows the original series to settle towards a limit. In our particular case, however, we determined it converges absolutely. But it's worthwhile to understand the distinction.

Sometimes, determining conditional convergence involves using the \( \text{Alternating Series Test} \), which considers criteria like:

• The magnitude of terms decreases (i.e., \( |a_{n+1}| \leq |a_n| \))
• Terms approach zero as \( n \) approaches infinity (i.e., \( \lim_{n\to\infty} a_n = 0 \))

These conditions must be met simultaneously for conditional convergence. Although our given series converges absolutely, understanding these principles will help in solving other series with just conditional convergence.