Question

An alternating series \(\sum_{n=i}^{\infty} a_{n}\) is given. (a) Determine if the series converges or diverges. (b) Determine if \(\sum_{n=0}^{\infty}\left|a_{n}\right|\) converges or diverges. (c) If \(\sum_{n=0}^{\infty} a_{n}\) converges, determine if the convergence is conditional or absolute. $$\sum_{n=0}^{\infty}(-1)^{n+1} \frac{3 n+5}{n^{2}-3 n+1}$$

Short answer

The series converges conditionally.

Step-by-step solution

Identify the series

The series given in the problem is an alternating series: \[ \sum_{n=0}^{\infty} (-1)^{n+1} \frac{3n+5}{n^2 - 3n + 1} \]An alternating series takes the form \( \sum_{n=0}^{\infty} (-1)^n b_n \) or \( \sum_{n=0}^{\infty} (-1)^{n+1} b_n \) where the terms alternate in sign. In this series, the terms alternate with respect to the expression \((-1)^{n+1}\).

Applying the Alternating Series Test

For the alternating series \( \sum_{n=0}^{\infty} (-1)^{n+1} a_n \), the series converges if the following two conditions are satisfied:1. \( \, |a_{n+1}| \leq |a_n| \) for all \( n \), meaning the sequence is non-increasing.2. \( \, \lim_{{n \to \infty}} a_n = 0 \).For our series, \( a_n = \frac{3n+5}{n^2-3n+1} \). Check if this sequence is non-increasing and approaches zero as \( n \to \infty \).

Check for decreasing sequence

We need to check whether \( a_n = \frac{3n+5}{n^2-3n+1} \) is non-increasing:1. Determine the expression for \( a_{n+1} = \frac{3(n+1)+5}{(n+1)^2 - 3(n+1) + 1} = \frac{3n + 8}{n^2 + 1} \).2. We compare \( a_n \) with \( a_{n+1} \) and find that as \( n \) increases, \( a_n \) gets smaller because the denominator grows faster compared to the numerator.3. Thus, the sequence \( a_n \) is indeed non-increasing.

Check limit as n approaches infinity

Find \( \lim_{{n \to \infty}} \frac{3n+5}{n^2-3n+1} \):- As \( n \to \infty \), the terms \( 3n \) in the numerator and \( n^2 \) in the denominator dominate:\[ \lim_{{n \to \infty}} \frac{3n+5}{n^2-3n+1} = \lim_{{n \to \infty}} \frac{3/n}{1 - 3/n + 1/n^2} = 0\]- Therefore, \( \lim_{{n \to \infty}} a_n = 0 \), confirming that the series satisfies the Alternating Series Test.

Conclude convergence via Alternating Series Test

Since both conditions of the Alternating Series Test are satisfied, the series \( \sum_{n=0}^{\infty}(-1)^{n+1} \frac{3n+5}{n^2-3n+1} \) converges.

Determine if series of absolute values converges

Now, consider the series of absolute values:\[ \sum_{n=0}^{\infty} \left| \frac{3n+5}{n^2-3n+1} \right| \]This is equivalent to the series \( \sum_{n=0}^{\infty} \frac{3n+5}{n^2-3n+1} \). Estimate the growth of the sequences:- The numerator grows linearly with \( 3n \) while the denominator grows quadratically with \( n^2 \).- The terms decrease as \( n \), but not fast enough to make the series converge because \( \frac{3n}{n^2} = \frac{3}{n} \) behaves like a harmonic series, which diverges.

Evaluate if convergence is conditional or absolute

Since \( \sum_{n=0}^{\infty} \left|a_n\right| \) diverges but \( \sum_{n=0}^{\infty} a_n \) converges, the original series \( \sum_{n=0}^{\infty}(-1)^{n+1} \frac{3n+5}{n^2-3n+1} \) converges conditionally. This means the series converges only when the sign is taken into account but diverges when considering absolute values.

Key concepts

Alternating Series Test

An alternating series is one where the signs of the terms alternate between positive and negative. The series can be written in the form like \[\sum_{n=0}^{\infty} (-1)^n b_n\] or \[\sum_{n=0}^{\infty} (-1)^{n+1} b_n\]. In such series, the terms flip signs as the index increases, which is evident in the expression \((-1)^{n+1}\).

The Alternating Series Test (AST) helps determine if an alternating series converges. For the series \(\sum_{n=0}^{\infty} (-1)^{n+1} a_n\), two main conditions must be satisfied:

• The sequence \(|a_{n+1}| \leq |a_n|\) should be true for all \(n\), indicating the sequence is non-increasing.
• The limit of the terms \(\lim_{{n \to \infty}} a_n = 0\).

If both conditions are met, then the series converges. For example, consider the alternating series given in the exercise, where \(a_n = \frac{3n+5}{n^2-3n+1}\). Checking it reveals:

• It is non-increasing, because the denominator grows faster than the numerator.
• The limit as \(n\) approaches infinity of \(a_n\) equals \(0\), as the degree of \(n\) in the denominator is higher.

By confirming these two conditions, we ensure this specific series converges per the AST guidelines.

Conditional Convergence

Convergence in a series does not necessarily mean it converges in all contexts. Conditional convergence occurs when a series \(\sum a_n\) converges, but the series of its absolute values \(\sum |a_n|\) does not. This situation implies that the convergence of the series is reliant on the alternating signs of its terms.

For example, in our exercise, after applying the Alternating Series Test, we know the alternating series converges. However, there is more to check. We inquire whether the series of absolute values \(\sum_{n=0}^{\infty} \left| \frac{3n+5}{n^2-3n+1} \right|\) converges. It is found that it diverges because the sequence \(\frac{3n+5}{n^2-3n+1} \), upon ignoring its sign, behaves like a harmonic series which is known to diverge.

Therefore, this series is conditionally convergent. It only converges when considering the alternating positive and negative signs. However, removing the signs causes it to diverge. Understanding conditional convergence helps in identifying when a series' convergence is deceptive and influenced by its nature of terms.

Absolute Convergence

A series is absolutely convergent if the series of absolute values \(\sum |a_n|\) converges. Absolute convergence is a stronger form of convergence than conditional convergence. If a series converges absolutely, it converges under any rearrangement of its terms. This is true because the absolute values ensure positive terms cannot cancel out their negative counterparts, which might lead to divergence.

In the given exercise problem, we examined whether the series of absolute values, \(\sum_{n=0}^{\infty} \left| \frac{3n+5}{n^2-3n+1} \right|\), converges. As this sequence resembles the divergent harmonic series \(\sum \frac{1}{n}\), it also diverges. Consequently, the original alternating series does not converge absolutely.

Recognizing whether a series converges absolutely or not provides essential insight into its behavior and stability. Absolute convergence guarantees that the total sum is invariant, no matter how the terms are ordered. If a series doesn’t meet this criterion, it might still converge conditionally, subject to the nature of rearrangements.