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. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{2^{n}}{n^{2}}$$

Short answer

The series \( \sum_{n=1}^{\infty} (-1)^{n} \frac{2^n}{n^2} \) diverges, both normally and absolutely.

Step-by-step solution

Identify the Series Components

The given series is \( \sum_{n=1}^{\infty} (-1)^{n} \frac{2^{n}}{n^{2}} \). This is an alternating series because of the \((-1)^{n}\) term, which causes the series terms to alternate between positive and negative.

Test for Alternating Series Convergence

To determine convergence of an alternating series \(\sum (-1)^{n}a_n\), we use the Alternating Series Test. The series converges if \( \lim_{n \to \infty} a_n = 0 \) and \( a_{n+1} \leq a_n \) for all \( n \). Here, \( a_n = \frac{2^{n}}{n^{2}} \). Since \( \lim_{n \to \infty} \frac{2^n}{n^2} eq 0 \) (it actually diverges to infinity), the series does not meet the convergence criteria.

Analyze Absolute Convergence

To check the absolute convergence, consider the series \( \sum_{n=1}^{\infty} \left| a_{n} \right| = \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} \). This is a geometric series with the dominant factor \(2^n\), which outweighs the denominator \(n^2\). Thus, the series diverges.

Determine the Type of Convergence

Since the original alternating series diverges due to the failure of the basic alternating series test, there is no need to determine whether the convergence is conditional or absolute. The series is divergent.

Key concepts

Convergence Tests

Convergence tests are essential tools in determining whether a series converges or diverges. In the context of an alternating series like \( \sum_{n=1}^{\infty} (-1)^{n} \frac{2^{n}}{n^{2}} \), one common method is the Alternating Series Test. This test helps us decide if a series converges based on two main criteria:

• The terms \( a_n \) must eventually decrease in a non-increasing order: \( a_{n+1} \leq a_n \).
• The limit of the terms must approach zero: \( \lim_{n \to \infty} a_n = 0 \).

If both conditions are satisfied, the series converges. Otherwise, it diverges. In our case, the series does not pass the test since \( \lim_{n \to \infty} \frac{2^n}{n^2} \) does not equal zero; instead, it grows infinitely.

Another vital class of tests used to analyze series are ratio and root tests, which help in assessing convergence by considering limits involving ratios or roots of successive terms. These tests, however, are more suited for series that might not alternate or need further analysis beyond the basic Alternating Series Test.

Understanding convergence tests is crucial as they allow mathematicians to handle different kinds of series that appear in various mathematical contexts.

Absolute Convergence

Absolute convergence refers to a series where the series of absolute values also converges. For the given series \( \sum_{n=1}^{\infty} (-1)^{n} \frac{2^{n}}{n^{2}} \), checking for absolute convergence requires looking at the series \( \sum_{n=1}^{\infty} \left| a_n \right| = \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} \).

If this altered series (where all terms are made positive) converges, the original series is absolutely convergent. This often implies that the series is well-behaved even if the terms are no longer alternated between positive and negative.

In the case of our example, \( \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}} \) is clearly a divergent series. This is because the geometric growth of \( 2^n \) overwhelms the polynomial damping effect of \( n^2 \), leading to unbounded growth. Hence, the original series does not converge absolutely.

Absolute convergence is a stronger condition than regular convergence and is always indicative of a more stable series.

Conditional Convergence

Conditional convergence is a fascinating concept that arises in series where the series converges, but does not absolutely converge. The condition is defined such that the alternating series test might pass, but because the positive term series diverges, the convergence is specific to the original alternating form.

• Conditional convergence occurs when \( \sum a_n \) converges, but \( \sum |a_n| \) diverges.
• It showcases the special balancing of alternating terms leading to a finite sum, even if their magnitudes individually suggest divergence.

In our exercise, we discover the series \( \sum_{n=1}^{\infty} (-1)^{n} \frac{2^{n}}{n^{2}} \) does not converge at all; hence, it doesn't meet the conditions for either absolute or conditional convergence.

It’s crucial to understand that conditional convergence implies that the order of terms is vital for maintaining convergence. Re-ordering can affect the sum, which highlights the delicate nature of such series. Note, however, that since our given series diverges completely, investigating whether it is conditional or absolute is unnecessary in this specific case.