Question

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sqrt{n}+1}{\sqrt{n}+3}\)

Short answer

The series does not converge at all.

Step-by-step solution

Understand the Series

The series is given by \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{\sqrt{n}+1}{\sqrt{n}+3}\). This is an alternating series, as it contains the factor \((-1)^{n+1}\), which causes the terms to alternate in sign.

Apply the Alternating Series Test

The alternating series test requires that the sequence \(a_n = \frac{\sqrt{n}+1}{\sqrt{n}+3}\) is positive, decreasing, and its limit as \(n\to\infty\) is zero. The term is positive and as \(n\) increases, the term \(\frac{\sqrt{n}+1}{\sqrt{n}+3}\) approaches 1. However, since it doesn't approach 0, the test fails.

Test for Absolute Convergence

To test for absolute convergence, consider the series \(\sum_{n=1}^{\infty} \left| \frac{\sqrt{n}+1}{\sqrt{n}+3} \right|\), which is essentially \(\sum_{n=1}^{\infty} \frac{\sqrt{n}+1}{\sqrt{n}+3}\). This resembles \(\sum_{n=1}^{\infty} 1\) for large \(n\), which does not converge as it is a harmonic series-like behavior.

Determine the Type of Convergence

Given that the alternating series test is inconclusive because the limit is not zero and the absolute series diverges, we conclude that the original series converges conditionally if it converges only as an alternating series. However, since it fails to meet the alternating series test conditions, the series diverges.

Key concepts

Alternating Series Test

An alternating series is a series where the terms alternate in sign, which is common in sequences involving factors like \((-1)^{n+1}\). When you apply the Alternating Series Test, there are specific conditions that need to be met for a series to converge:

• The sequence \(a_n\) must be positive for all \(n\).
• The sequence \(a_n\) must be monotonically decreasing (i.e., each term must be smaller than the previous one).
• The limit of \(a_n\) as \(n o \infty\) must be zero.

For our series \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{\sqrt{n}+1}{\sqrt{n}+3}\), while each term is positive, it fails the test because the limit of \(\frac{\sqrt{n}+1}{\sqrt{n}+3}\) as \(n o \infty\) approaches 1, not 0. This oversight indicates that the series does not converge by the alternating series criteria.

Absolute Convergence

Absolute convergence in a series happens when the series of absolute values converges. To test for this, you take the absolute value of each term, making sure all terms are positive, and observe whether this new series converges.

• If \(\sum |a_n|\) converges, then the original series \(\sum a_n\) converges absolutely.
• If \(\sum |a_n|\) diverges, the original series could still potentially converge conditionally or not converge at all.

For our example, we look at the series \([\sum_{n=1}^{\infty} \frac{\sqrt{n}+1}{\sqrt{n}+3}]\). It bears a resemblance to the harmonic series, which is known to diverge. This means that our series does not converge absolutely since the absolute version of the series exhibits divergence.

Conditional Convergence

A series converges conditionally if it converges when considered as an alternating series but does not converge absolutely. It's a special type of convergence that occurs in alternating series, typically found when absolute values of terms lead to a divergent series.

• This happens when \(\sum a_n\) converges, but \(\sum |a_n|\) does not.
• For conditional convergence, the alternating sign's effect on the sum is crucial.

In the case of our series \([\sum_{n=1}^{\infty} (-1)^{n+1} \frac{\sqrt{n}+1}{\sqrt{n}+3}]\), because it neither passes the alternating series test (as the limit does not go to zero) nor converges absolutely, we determine it does not converge at all. Conditional convergence would require the alternating nature to still allow the series to hold, which it does not in this case.