Question

Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} a_{n} \text { where } a_{n}=\frac{1}{n(n+1 / 2)} $$

Short answer

The series converges by the comparison test.

Step-by-step solution

Identify the Series

The series given is \[\sum_{n=1}^{\infty} \frac{1}{n(n+1/2)}.\] Our goal is to determine whether this series converges.

Find a Comparison Series

To use the comparison test, select a simpler series for comparison. Notice that \[a_n = \frac{1}{n(n + 1/2)} < \frac{1}{n^2}\]for all \(n\). Thus, we can compare \(a_n\) with the convergent series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\), oting that \(\frac{1}{n^2}\) is a well-known convergent p-series with \(p = 2 > 1\).

Apply the Comparison Test

Since \(0 \leq a_n = \frac{1}{n(n+1/2)} < \frac{1}{n^2}\) and the series \(\sum \frac{1}{n^2}\) converges, by the comparison test, the series \(\sum_{n=1}^{\infty} a_n\) converges.

Conclusion

Using the comparison test with the convergent p-series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\), we determined that the series\[\sum_{n=1}^{\infty} \frac{1}{n(n+1/2)}\]also converges.

Key concepts

Convergence of Series

In mathematics, a series is the sum of the terms of a sequence. When we discuss the "convergence of a series," we're talking about whether the sum of the sequence's terms approaches a specific value as more and more terms are added. If it does, we say the series is convergent.

To better understand convergence, consider an infinite series \( \sum_{n=1}^{\infty} a_n \). For this series to be convergent, the sequence of partial sums, \( S_N = a_1 + a_2 + ... + a_N \), must approach a finite limit as \( N \) approaches infinity.

Several tests can help determine if a series converges, such as the

• Comparison Test: Compares the series with a known convergent (or divergent) series to make conclusions.
• Ratio Test: Uses the ratio of successive terms to determine convergence.
• Root Test: Utilizes the nth-root of terms to analyze convergence.

Recognizing the type of series you are dealing with is crucial to decide which test to apply.

P-Series

A p-series is a special type of series that takes the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive constant. The convergence or divergence of a p-series depends on the value of \( p \):

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

For example, the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), where \( p=2 \), is convergent because \( p > 1 \). This is a commonly referenced case of a convergent p-series and often serves as the basis for comparison in other tests, such as the comparison test. Understanding p-series helps immensely in identifying convergence in more complex series, making them a fundamental concept in calculus.

Comparison Test for Convergence

The comparison test is a robust tool for determining the convergence or divergence of an infinite series. This test involves comparing a series whose convergence is unknown with another series whose behavior is already known.

If \( 0 \leq a_n \leq b_n \) for all \( n \) beyond some initial index, and \( \sum b_n \) is a known convergent series, then \( \sum a_n \) must also converge. In contrast, if \( \sum b_n \) is known to diverge, and \( a_n \) is always greater or equal to \( b_n \), then \( \sum a_n \) also diverges.

This test is particularly useful when dealing with series that are complicated or difficult to analyze on their own, like the original series \( \sum_{n=1}^{\infty} \frac{1}{n(n+1/2)} \). By comparing it to the simpler convergent series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), we conclude the unknown series converges, due to its terms being smaller than the well-known convergent p-series with \( p = 2 \). This approach simplifies complex problems by breaking them down into familiar and manageable pieces.